DESIGN  AND  CONSTRUCTION 

OF 

HEAT  ENGINES 


PUBLISHERS     OF     BOOKS 

Coal  Age     ^     Electric  Railway  Journal 

Electrical  World  *  Engineering  News-Record 

American  Machinist  v  Ingenierfa  Internacional 

Engineering 8 Mining  Journal      ^     Power 

Chemical  S   Metallurgical  Engineering 

Electrical  Merchandising 


DESIGN  AND  CONSTRUCTION 

OF 

HEAT  ENGINES 


BY 

WM.  E.  NINDE,  M.  E. 

ASSOCIATE   PROFESSOR   OP   MECHANICAL  ENGINEERING,   SYRACUSE   UNIVERSITY 
MEMBER   OP   AMERICAN   SOCIETY   OF   MECHANICAL   ENGINEERS 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    239  WEST  39TH  STREET 

LONDON:    6  &  8  BOUVERIE  ST.,  E.  C.  4 

1920 


Engineering? 
Library 


COPYRIGHT,  1920,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE    MAPLE    PKE  S  S    YORK 


DEDICATION 

To  the  man  who  esteems  excellence 

in  his  calling  of  greater  value 

than  his  remuneration 


414821 


PREFACE 

The  object  of  this  book  is  to  supply  in  one  volume  the  material  most 
essential  to  the  well-equipped,  independent  designer  of  heat  engines,  and 
to  give  this  material  in  the  form  most  convenient  for  use  in  class  room  and 
practical  work,  by  a  separate  treatment  of  the  different  phases  of  the 
subject. 

Contributory  to  the  contents  are  the  author's  note  books  on  engine 
design  covering  his  practice  and  observation  for  over  twenty  years; 
revisions  and  additions  necessary  to  adapt  the  notes  to  his  teaching  work 
for  the  past  ten  years;  material  from  the  best  technical  books  and  peri- 
odicals necessary  to  fill  the  gaps  in  first-hand  information,  and  to  add 
breadth  and  character  to  the  work;  and  data  and  drawings  from  some 
of  the  best  designers  and  builders  of  heat  engines  in  the  United  States- 
giving  a  practical,  commercial  touch. 

In  the  derivation  of  working  formulas,  elaborate  and  abstruse  methods 
have  been  avoided  as  much  as  possible,  but  no  important  element  which 
permits  of  practical  treatment  has  been  omitted;  also,  while  reverting 
to  fundamental  principles  and  making  the  formulas  as  rational  as  possible, 
their  limitations  are  pointed  out,  and  they  are  usually  brought  to  a  form 
suitable  for  direct  application. 

Illustrations  of  none  but  the  most  excellent  designs  are  provided. 
These  are  necessarily  limited  on  account  of  space,  but  qualitative  design, 
and  certain  details  and  auxiliaries  may  be  studied  in  books  of  a  more 
descriptive  character,  and  in  builders'  catalogs  which  are  always 
obtainable. 

The  numbering  of  formulas  begins  with  each  chapter,  and  manu- 
facturers' material  is  usually  placed  in  the  latter  part  of  the  chapter  to 
facilitate  the  revisions  necessary  to  keep  the  work  up  to  date.  The  nota- 
tion for  a  chapter  is  listed  at  its  beginning  except  for  short,  scattered 
discussions,  when  it  is  given  only  in  the  text. 

Steam  tables  and  other  tables  found  in  all  handbooks  are  omitted; 
they  are  more  convenient  to  use  with  the  formulas  of  this  book  if  in  a 
separate  cover. 

A  few  references  are  given  at  the  ends  of  chapters  to  extend  the  scope 
of  the  work. 

vii 


viii  PREFACE 

The  process  of  absorption  by  contact  with  associates  in  office,  shop 
and  college,  and  habitual  gleaning  from  technical  periodicals,  books  and 
catalogs  long  before  the  writing  of  a  book  was  thought  of,  renders  the 
placing  of  credit  difficult,  but  when  possible  it  has  been  done. 

The  author  is  very  grateful  to  the  manufacturers  and  engineers  who 
have  furnished  illustrations  and  data,  and  to  authors  and  publishers  who 
have  permitted  the  use  of  material  from  their  publications.  Thanks  are 
also  due  my  wife,  Luella  V.  Ninde,  whose  assistance  and  encouragement 
have  made  the  book  possible. 

W.  E.  N. 

SYRACUSE  UNIVERSITY, 
January,  1920. 


CONTENTS 

PAGE 
PREFACE *   .   .   i vii 

Part  1.  The  Heat  Engine 

CHAPTER  I 

STATUS  OP  THE  HEAT  ENGINE    .....;. 1 

Furnishes  bulk  of  power — Water  power  inadequate — Adaptability  of  dif- 
ferent types. 

CHAPTER  II 

THE  POWER  PLANT '3 

Steam  cycle  not  complete  in  engine — Cycle  of  steam  operation — Internal- 
combustion  engine. 

CHAPTER  III 

THE  STEAM  ENGINE 5 

Mechanism — Principle  of  operation — Function  of  valve  gear — Indicator 
diagrams — Types  and  classification — The  uniflow  engine. 

CHAPTER  IV 

THE  STEAM  TURBINE 23 

Of  early  origin — Impulse  and  reaction — Commercial  classification — Simple 
impulse  turbine — Compound  impulse  turbine — Velocity  stage — Pressure 
stage — Reaction  turbine — Combinations — Governing 

CHAPTER  V 

THE  INTERNAL-COMBUSTION  ENGINE 32 

Classification  and  cycles — Gas  engines — Light-oil  engines — Heavy-oil 
engines — The  Diesel  engine — Governing — Starting — Cylinder  cooling  and 
mufflers — Practical  data. 

Part  2.  Thermodynamics 
CHAPTER  VI 

GENERAL  POWER  FORMULAS  AND  GASES 51 

Power  of  heat  engines — Mean  effective  pressure — Mechanical  efficiency — 
Practical  cycles  for  using  gas — The  constant-volume  cycle — The  constant- 

ix 


X  CONTENTS 

PAGE 

pressure  cycle — Volumetric  efficiency — Temperature  rise  due  to  combustion 
— Conventional  indicator  diagrams. 

CHAPTER  VII 

STEAM .    ;    .    . 65 

Formation  under  constant  pressure — Relation  of  pressure  and  temperature — 
Supply  of  heat  during  formation — Flow  of  steam — Effect  of  friction  on  flow. 

CHAPTER  VIII 

ECONOMY 70 

The  steam  plant — The  prime  mover — I.h.p.  and  b.h.p. — The  heat  factor — 
Internal-combustion  engines — Comparative  economy. 

CHAPTER  IX 

CYLINDER  EFFICIENCY 78 

The  steam  engine  cylinder — Condensation  and  re-evaporation — Compression 
and  clearance — Factors  affecting  cylinder  condensation — The  internal- 
combustion  engine  cylinder. 


Part  3.  Friction  and  Lubrication 
CHAPTER  X 

MECHANICAL  EFFICIENCY 97 

Friction  horsepower — Effect  of  different  methods  of  lubrication — Variation 
of  efficiency  with  load — Effect  of  maximum  steam  or  gas  pressure. 

CHAPTER  XI 

LUBRICATION 103 

General  principles — Lubricants  and  their  application — Clearance — Oil 
grooves — Bearing  metal — Bearing  proportions — Thrust  bearings — Eccentric 
bearing — Force-feed  systems  of  lubrication — Cylinder  lubrication. 


Part  4.  Power  and  Thrust 
CHAPTER  XII 

THE  SIMPLE  STEAM  ENGINE 127 

Indicator  diagrams — Mean  effective  pressure — Diagram  factors — Governing 
— Piston  thrust — Indicated  horsepower — Theoretical  steam  consumption — 
Compression — Standard  engines — Application  of  formulas. 


CONTENTS  xi 

PAGE 

CHAPTER  XIII 

THE  COMPOUND  STEAM  ENGINE 152 

Indicator  diagrams — Condensing  engines — Influence  of  the  receiver — The 
tandem  compound— The  cross  compound — Governing — Maximum  thrust — 
Indicated  horsepower — Diagram  factors — Theoretical  steam  consumption — 
Standard  engines — Application  of  formulas. 

CHAPTER  XIV 

THE  INTERNAL-COMBUSTION  ENGINE 185 

Mean  effective  pressure — Horsepower — Practical  constants — Simple 
formulas — Effect  of  various  factors  upon  capacity — Compression — Govern- 
ing— The  quality  method — The  quatity  method — Combined  methods — 
Rating — Indicator  diagrams  and  maximum  piston  thrust — Maximum  indi- 
cator diagram — Application  of  formulas. 

CHAPTER  XV 

THE  STEAM  TURBINE 206 

The  general  method — Nozzles  and  other  passages — Effect  of  friction — 
Practical  notes  on  nozzles  and  other  passages — Impulse  blading — Practical 
notes  on  impulse  blading — Velocity-stage  impulse  turbine — Practical  notes 
on  velocity-stage  turbine — Intermediate  pressures— Effect  of  heat  factor 
— Peabody's  direct  method — Superheated  steam — Velocity  due  to  reheating 
— General  dimensions — Number  of  equal  stages — Unequal  wheel  diameters — 
Conservation  of  residual  velocity — Pressure-velocity-stage  impulse  turbine — 
Trajectory  and  lead — The  reaction  turbine — Reaction  blading— Work 
division — Practical  notes  on  reaction  turbines — Factors  influencing  turbine 
operation — The  condenser — Steam  passages — Governing — Rating  and  over- 
load— Application  of  formulas. 

Part  5.  Mechanics 
CHAPTER  XVI 

THE  SLIDER  CRANK  . 283 

Properties  of  the  connecting  rod — Piston  velocity — Forces  due  to  pressure 
in  cylinder — Efficiency  of  the  slider  crank — Forces  due  to  gravity — Forces 
due  to  acceleration — The  reciprocating  parts — The  connecting  rod — The 
effect  of  angular  acceleration — Reaction  on  the  guide — Combined  indicator 
and  inertia  diagram — Steam  engine — Internal-combustion  engine — Reversal 
of  thrust  and  effect  of  compression — Total  turning  effort — Combined  turning 
effort  diagrams — Reaction  on  the  frame — Application  to  steam  and  internal- 
combustion  engines. 

CHAPTER  XVII 

BALANCING 334 

Simple,  or  primary  balancing — Reciprocating  parts — Secondary  balance — 
Multi  -  cylinder  engines — The  connecting  rod — Turning  effort — Balance 
weights. 


Xll  CONTENTS 

PAGE 
CHAPTER  XVIII 

REGULATION   DURING  THE  CYCLE.     FLYWHEELS 350 

The  control  of  speed  fluctuation — The  control  of  displacement — Com- 
parison of  methods — Application  to  practice. 

CHAPTER  XIX 

REGULATION  DURING  CHANGE  OF  LOAD.     GOVERNORS 367 

Independent  method — Coordinate  method — Governor  types — Centrifugal 
governors — Stability — Isochronism — Inertia,  or  centrifugal  -  inertia  gover- 
nors— Relay  governors — Safety,  or  over  speed  governors — Dashpots — Speed 
adjustments — Speed  fluctuation — Speed  variation — General  equations  for 
centrifugal  governors — Equations  for  conical  governors — Determination  of 
speed — Equations  for  spring  governors — Springs — Practical  considerations — 
Gravity  balance — Constants  of  regulation — Machine  design — Illustrations 
from  practice. 

CHAPTER  XX 

VALVES  AND  VALVE  GEARS 401 

Port  area — Allowable  fluid  velocity — Classification — The  single-valve  gear — 
Lap,  lead  and  angular  advance — Effect  of  rod  angularity — The  Bilgram  dia- 
gram— Double-ported  valves — Shifting  and  swinging  eccentrics — Rectangular 
valve  diagrams — Corliss  releasing  gear — Releasing  gear  operation — High- 
speed Corliss  gears — Mclntosh  and  Seymour  gear — Cam  -  and  -  eccentric 
gears — Reversing  gears  for  steam  engines — Equivalent  eccentric  for  Stephen- 
son  and  Walschsert  gears — Variable  lead  for  Walschsert  gear — Hackworth 
gear —  Marshall  gear — Joy  gear — Doble  gear — Internal-combustion  engine 
gears — Multi-cylinder  gears — Eccentric-and-cam  gears — Valve  springs — 
Two-cycle  engines — Reversing  gears  for  internal-combustion  engines — The 
sleeve  motor  gear — Details  from  practice. 

Part  6.     Machine  Design 
CHAPTER  XXI 

GENERAL  CONSIDERATIONS 463 

Rational  machine  design — Working  stresses  and  factors  of  safety — Static 
stress — Repeated    stress — Reversed    stress — Suddenly   applied   load    and 
shock — Mechanical  properties  of  materials — Selection  of  formulas — Cylinder 
walls — Combined  bending  and  twisting — Rectangular  section  in  torsion — 
Struts — Clearances  and  tolerances — Basis  of  design. 

CHAPTER  XXII 

CYLINDERS 499 

The  cylinder — Counterbore — Inlet  and  exhaust  passages — Cylinder  heads — 
Stuffing  box — Cylinder  lagging — Designs  from  practice. 


CONTENTS  xiii 

PAGE 

CHAPTER  XXIII 

PISTONS 514 

Material — Strength  calculations — Piston  rings — Eccentric  rings — Rings  of 

uniform  section — Number  of  rings — Designs  from  practice. 

• 

CHAPTER  XXIV 

PISTON  RODS .'•  .'', 530 

Piston  rod  formulas — The  crosshead  end — The  piston  end — Numerical 
values — The  piston  rod  as  a  strut — Pressed  fits — Designs  from  practice. 

CHAPTER  XXV 

CONNECTING  RODS 544 

Body  of  rod — A  general  formula  for  direct  calculation  of  dimensions — 
Numerical  values — Circular  section — Rectangular  section — I  section — Con- 
necting rod  ends — Wedge  bolts — Designs  from  practice. 

CHAPTER  XXVI 

CROSSHEADS 562 

The  crosshead — Crosshead  pin — Crosshead  shoe — Application  of  formulas — 
Designs  from  practice. 

CHAPTER  XXVII 

CRANKS ,   . 572 

Crank  pin — Crank  arm — Hubs — Crank  designs. 

CHAPTER  XXVIII 

SHAFTS 586 

Shaft  types — The  side-crank  shaft — The  center-crank  shaft — Methods  of 
calculation — General  formulas — Special  formulas — Application  of  formulas 
to  steam  engine  and  gas  engine  cranks— Designs  from  practice. 

CHAPTER  XXIX 

FRAMES 611 

Stresses  in  engine  frames — Side-crank  frames — Main  bearings — Wedge  ad- 
justing bolts — Cap  bolts — Designs  from  practice. 

CHAPTER  XXX 

FLYWHEELS 623 

Hoop  stress — Unwin's  formulas — Unwin's  method  modified — Forces  pro- 
ducing bending  stresses  in  arms — Number  of  arms  carrying  load — Bending 


XIV  CONTENTS 

PAGE 

moment  in  arms — Working  stresses — Rim  and  arm  sections — Hubs — Hub 
bolts — Shear  bolts — Rim  bolts  and  links — Keys — Methods  of  construction — 
Application  of  formulas — Designs  from  practice. 

*    CHAPTER  XXXI 

TURBINE  WHEELS. .4    .........   649 

Stresses  in  wheels — Material — Blading  design — Application  of  formulas — 
Designs  from  practice. 

CHAPTER  XXXII 

TURBINE  SHAFTS .664 

Nature  of  the  problem — Critical  speed — Deflection — Application  of 
formulas. 

CHAPTER  XXXIII 

TURBINE  CASINGS  AND  DETAILS 674 

DeLaval  turbine — Ridgway  turbine. 

CHAPTER  XXXIV 

GENERAL  ARRANGEMENTS  AND  FOUNDATIONS 679 

General  arrangements — Foundations — Vibration  problems — Material — 
Bolts  and  washers — Construction — Grouting — The  soil — Specification  for 
foundation. 

CHAPTER  XXXV 

DESIGN  METHODS ......................   687 

Personal  efficiency — Order — Thoroughness — Study — The  slide  rule — Stand- 
ards— Partial-assembly  sketches. 

APPENDIX  1 
MOMENT  OP  INERTIA  OF  IRREGULAR  SECTIONS  . »   ,    .    .  693 

APPENDIX  2 

UNITED  STATES  STANDARD  BOLTS  AND  NUTS 695 

INDEX.  .   697 


DESIGN  AND  CONSTRUCTION 

OF 

HEAT  ENGINES 

PART  I-THE  HEAT  ENGINE 


CHAPTER  I 
STATUS  OF  THE  HEAT  ENGINE 

It  is  roughly  estimated  that  85  per  cent,  of  the  power  utilized  in  the 
United  States  for  manufacturing  plants,  central  power  plants  and  elec- 
tric-railway power  stations  is  furnished  by  heat  engines,  the  remaining 
15  per  cent,  being  water  power.  This  does  not  include  locomotives, 
steamships  or  any  form  of  self-propelled  vehicles,  which  would  increase 
the  heat-engine  percentage  still  more. 

It  has  also  been  estimated  that  the  total  available  water  power  in  the 
United  States  approximates  the  present  output  of  steam  power.  How- 
ever, by  the  time  this  power  is  developed  it  is  probable  that  the  demand 
will  have  so  increased  that  the  power  developed  by  the  heat  engine  in 
its  varied  forms  will  be  as  greatly  in  excess  as  at  the  present  time.  The 
importance  of  heat  engines  is  then  obvious  and  their  manufacture  will 
continue  to  be  an  important  industry  until  our  fuel  supply  is  exhausted 
or  some  radical  discovery  displaces  them. 

Heat  engines  may  be  divided  into  three  important  classes  which  are 
commercially  successful  today,  viz.:  1.  The  steam  engine;  2,  the  steam 
turbine;  and  3,  the  internal-combustion  engine.  The  last  named  may 
properly  be  divided  into  two  commercial  forms:  (a)  The  gas  engine  and 
(6)  the  oil  engine,  the  former  using  gaseous,  the  latter  liquid  fuel.  The 
gas  turbine  is  not  yet  recognized  in  the  commercial  field  and  will  not  be 
further  mentioned. 

Heat  engines  may  also  be  classified  as:  1.  Prime  movers;  and  2, 
reversed  heat  engines,  such  as  the  compression  refrigerating  machine  and 
the  air  compressor.  Prime  movers  only  will  be  considered. 

The  relative  importance  of  the  different  classes  is  an  open  question 

1 


2  DESIGN  AND  COti&PtfUCTlON  OF  HEAT  ENGINES 

but  it  is  probable  that  each  has  its  particular  field  of  usefulness  which  in 
some  cases  cannot  be  as  well  provided  for  by  the  others. 

Statements  were  current  in  the  technical  press  a  few  years  ago  that 
little  was  to  be  expected  in  the  way  of  improved  economy  for  the  recip- 
rocating steam  engine,  and  its  passing  was  predicted.  That  the  limit 
of  improvement  had  been  nearly  reached  may,  at  that  time  have  been 
true  across  the  sea,  but  since  then  in  this  country,  the  application  of 
superheated  steam,  the  uniflow  principle  and  the  locomobile,  together 
with  improvements  in  materials  and  some  details  of  construction  which 
have  made  possible  higher  piston  and  rotative  speeds  in  the  larger  units, 
have  greatly  improved  the  steam  engine,  so  that  it  is  holding  its  own  in 
all-around  efficiency  and  reliability  with  other  forms.  The  steam  loco- 
motive, that  once  proverbially  wasteful  machine,  now  equals  in  economic 
performance  some  of  the  best  stationary  steam  plants  of  a  few  years  ago. 

But  the  reciprocating  engine  has  limits  in  the  size  of  its  units,  and 
beyond  this  the  steam  turbine  is  best  adapted.  The  turbine,  with  its 
high  rotative  speed  and  uniform  rotation  is  well  fitted  to  drive  alternating- 
current  generators  of  large  capacity.  The  floor  space  is  also  less  than  for 
a  reciprocating  engine  of  the  same  power. 

The  internal-combustion  engine  has  attained  the  highest  efficiency  of 
all  heat  engines,  and  this  is  probably  responsible  for  the  prediction  that 
the  days  of  the  steam  engine  were  numbered.  Notwithstanding  all 
this,  the  internal-combustion  engine  furnishes  at  the  present  time  but 
a  small  percentage  of  the  output  of  stationary  power  plants,  but  there 
seems  little  doubt  but  that  this  ratio  will  be  increased.  It  is  a  reciprocat- 
ing engine,  and  in  this  particular  may  be  classed  with  the  steam  engine. 

Power,  in  the  issue  of  Nov.  17,  1914,  contains  the  statement  that  the 
total  power  of  automobile  engines  manufactured  the  previous  year  was 
equivalent  to  twice  the  potential  power  of  Niagara  Falls,  or  13,500,000 
horsepower.  This,  coupled  with  the  fact  that,  notwithstanding  the 
increasing  demands  for  large  turbine  units,  the  reciprocating  engine  is 
still  largely  in  excess,  makes  it  probable  that  the  passing  of  the  reciprocat- 
ing engine  is  not  imminent. 

It  is  not  intended  to  compare  the  merits  of  the  different  types  of 
prime  movers,  or  the  different  designs  of  a  given  type,  as  practically 
all  when  well  designed,  constructed  and  operated  give  good  results  in 
their  respective  fields.  The  determination  of  the  best  machine  for  a  given 
set  of  conditions  is  often  much  involved  and  is  outside  the  scope  of  this 
work.  It  is  undoubtedly  true  that  the  periodic  popularity  of  the  various 
machines  is  not  always  based  upon  a  profound  knowledge  of  their  actual 
merits. 


CHAPTER  II 
THE  POWER  PLANT 

The  Power  Plant  includes  the  prime  mover  and  all  other  appliances 
and  accessories  necessary  for  the  production  of  power,  for  manufacturing 
plants,  central  stations,  ship  propulsion,  locomotives  and  automobiles,  or 
for  any  form  of  activity  requiring  power.  The  development  of  the  power 
plant  has  been  such  that  the  selection  and  arrangement  of  its  various 
apparatus  has  become  a  highly  specialized  profession.  The  work  of  the 
power-plant  designer  and  of  the  designers  of  the  plant  apparatus  is  inter- 
dependent, the  latter  making  possible  great  advancement  in  plant 
efficiency,  while  on  the  other  hand  the  demands  of  the  plant  designer 
have  been  an  incentive  to  the  designer  of  power  apparatus. 

Power-plant  design  is  not  within  the  range  of  this  book,  and  as  the 
development  of  power  apparatus  is  in  a  state  of  continual  progress,  de- 
scriptions of  different  designs  will  not  be  attempted,  the  task  of  keeping 
pace  with  the  substantial  progress  in  heat-engine  design  being  thought 
sufficient;  however,  the  steam  cycle  is  not  completed  in  the  engine  or 
turbine,  and  in  view  of  future  references  to  the  different  phases  of  the 
cycle,  the  path  of  the  steam  through  the  plant  will  be  traced. 

Steam  is  generated  in  the  boiler  by  the  combustion  of  fuel  in  the  boiler 
furnace.  From  thence  it  is  conducted  through  piping  to  the  turbine  or 
engine  where  it  does  work.  It  is  then  exhausted,  sometimes  into  the 
atmosphere,  which  is  very  wasteful;  sometimes  into  pipes  for  heating 
purposes  or  for  some  industrial  process,  and  it  is  often  exhausted  into  a 
condenser.  There  it  comes  into  contact  directly,  as  in  the  jet  condenser, 
or  indirectly,  as  in  the  surface  condenser,  with  cold  water.  The  heat  is 
withdrawn  from  the  steam  and  it  condenses  back  to  water,  and  due  to  the 
great  difference  between  the  volume  of  a  given  weight  of  water  and  of 
steam,  a  partial  vacuum  is  formed,  often  approaching  within  three-quarters 
of  a  pound  of  a  perfect  vacuum.  With  either  type  of  condenser  the 
water  is  pumped  out,  together  with  any  air  which  entered  with  the  steam 
or  water. 

Water  must  be  pumped  into  the  boiler,  either  by  the  boiler  feed 
pump  or  the  injector,  to  replace  the  water  used  by  the  engine,  and  this 
completes  the  cycle.  In  most  large  plants  the  water  is  passed  through 
a  feed-water  heater  before  going  to  the  boiler.  This  is  heated  by  the 

3 


4  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

exhaust  from  the  auxiliaries  or  main  engine,  or  sometimes  by  the  hot 
gases  of  the  flue  leading  from  the  furnace  to  the  chimney;  in  the  latter 
case  it  is  called  an  economizer  by  common  usage. 

If  a  surface  condenser  is  used,  the  condensed  steam  may  be  pumped 
back  into  the  boiler,  and  barring  leakage  losses,  a  nearly  continuous  cycle 
with  the  same  water  may  be  effected,  approaching  the  theoretical  cycle. 

With  the  internal-combustion  engine  the  thermal  cycle  is  complete 
in  the  engine  cylinder.  There  are,  however,  numerous  accessories.  For 
the  gas  engine  there  must  be  the  gas  producer,  and  the  scrubbers  for 
cleaning  the  gas;  with  the  oil  engine,  apparatus  for  handling  the  oil  must 
be  provided;  air  compressors  are  also  necessary  for  some  types  and  vapor- 
izers or  carbureters  for  others.  These  may  be  considered  as  part  of  the 
power-plant  equipment,  but  are  not  part  of  the  engine  proper,  although  in 
many  cases  they  are  furnished  by  the  engine  builder. 


CHAPTER  III 
THE  STEAM  ENGINE 

1.  The  Steam  Engine. — The  history  of  the  steam  engine  is  full  of 
interest,  and  should  form  a  part  of  the  reading  of  every  steam  engineer. 
In  the  popular  mind  the  profession  of  mechanical  engineering  is  closely 
associated  with  the  steam  engine,  and  in  fact  owes  much  to  it  as  an  impor- 
tant factor  in  the  separation  of  this  branch  of  science  from  the  general 
field  of  engineering,  or  civil  engineering,  which  formerly  included  the 
several  now  distinct  branches. 

In  order  to  reserve  space  for  present-day  practice,  it  was  thought  best 
to  refer  to  the  works  of  Rankine,  Thurston,  Ewing  and  others,  and  to 
eliminate  from  this  text  practically  all  historical  matter. 

2.  Mechanism. — With  the  exception  of  comparatively  small  pumps, 
all  practical  reciprocating  engines  employ   for  the  conversion  of  heat 
energy  into  useful  work,  the  simple  slider-crank  mechanism.     This  mech- 
anism also  resolves  reciprocating  into  rotary  motion,  or  vice  versa  in  the 
case  of  reversed  heat  engines  such  as  compressors.     For  engine  purposes 
the  slider-crank  mechanism  is  the  simplest  and  most  economical  from  the 
standpoint  of  both  construction  and  operation,  and  although  attempts 
have  been  made  to  improve  upon  it,  they  have  been  fruitless.     Neglecting 
friction,  which  in  some  cases  is  as  low  as  4  per  cent,  of  the  entire  power 
developed  in  the  cylinder,  the  efficiency  is  100  per  cent.,  as  proven  in  Par. 
100,  Chap.  XVI. 

A  diagram  of  the  slider-crank  mechanism  as  applied  to  the  steam 
engine  is  shown  in  Fig.  1.  The  mechanism  proper  consists  of  the  frame 
(/),  the  crank  (e),  the  connecting  rod  (d)  and  the  crosshead  (c), 
sliding  upon  the  guides  which  restrict  its  movement  to  a  straight  line. 
The  piston  (a)  and  piston  rod  (6)  may  be  considered  as  an  extension  to  the 
crosshead  and  serve  to  transmit  the  pressure  of  the  steam  to  it. 

Ports  for  the  admission  and  outlet  of  steam  are  shown  at  1,  2,  3  and  4. 
In  some  engines  the  same  ports  are  used  for  inlet  and  outlet.  The  con- 
trol of  the  steam  passing  into  and  from  the  cylinder  is  effected  by  valves 
which  are  actuated  by  an  eccentric,  a  form  of  crank,  which  is  located  on 
the  engine  shaft.  In  Fig.  1,  valves  1  and  2  are  inlet 'valves,  while  3  and 
4  are  outlet  or  exhaust  valves. 

To  further  illustrate  the  action  of  the  steam  engine  as  a  whole,  Figs. 

5 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


2  to  9  are  given.  Contrary  to  the  usual  practice  of  introducing  valve 
action  with  a,  single-valve  engine,  it  is  thought  that  less  confusion  will 
accompany  the  conception  of  a  separate  valve  controlling  each  operation 
concerned  with  the  handling  of  steam  during  the  engine  cycle,  which 


FIG.  1 


with  the  steam  engine  is  effected  in  one  revolution  of  the  crank;  conse- 
quently, a  four-valve  engine  of  the  nonreleasing  type  is  shown. 

Figure  2  is  the  side  of  the  engine  showing  the  crank  and  connecting 
rod.     Figure  3  shows  the  valve-gear  side  with  the  flywheel  removed. 


1? 

Qb 

^y 

0 

'^~ 

OJ 

FIG.  2. 


Figure  4  shows  the  valve-gear  side  with  the  cylinder  in  section,  the  rods 
and  levers  operating  the  gear  being  shown  by  dotted  lines,  as  are  also 
the  crank  and  connecting  rod. 

Figures  5,  and  7  to  9  show  valves,  with  center  lines  only  of  wrist 


FIG.  3. 


plate  arms,  levers,  crank  and  eccentric,  in  order  to  show  the  different 
positions  more  clearly.  This  forms  a  valve  diagram,  treated  more  fully 
in  Chap.  XX. 

In  practice,  with  this  type  of  gear,  the  eccentric  changes  position  in 
relation  to  the  crank,  under  the  control  of  the  governor  located  on  the 


THE  STEAM  ENGINE 


shaft,  as  the  load  on  the  engine  Is  varied,  but  for  simplicity,  this  is  neg- 
lected in  the  present  treatment,  and  the  eccentric  is  assumed  in  a  fixed 
position  on  the  shaft.     All  of  these  details, 
and  the  exact  action  of  the  valves,  are  fully 
treated  in  Chap.  XX. 

In  Fig.  3,  (a)  is  the  eccentric,  which  by 
means  of  the  eccentric  strap  (6)  and  eccentric 
rod  (c)  connects  with  the  rocker  or  carrier 
(d).  The  reach  rod  (e)  connects  to  the  wrist 
plate  (/) .  Links  (</)  connect  the  wrist  plate  to 
the  valve  levers  (h)  and  (k) . 

It  is  plainly  seen  that  as  the  crank  revolves, 
a  rocking  motion  is  given  to  the  wrist  plate, 
which  in  turn  causes  the  valve  levers  to 
oscillate  through  a  given  angle.  The  valve 
levers  being  keyed  to  the  valve  stems,  impart 
motion  to  the  valves,  opening  and  closing  the 
ports,  thus  controlling  the  steam  supply  to  the 
cylinder. 

The  valve  gear  is  shown  in  its  central 
position,  and  the  crank  is  at  the  head-end  dead 
center.  When  the  valves  are  properly  set  this 
would  not  be  the  case;  the  eccentric  would 
be  advanced  for  this  position  of  the  crank 
as  shown  in  Fig.  4. 

Figure  4  is  a  sectional  view  of  the  same 
engine.  The  end  of  the  cylinder  toward  the 
crank  is  called  the  crank  end,  and  the  opposite 
end  the  head  end.  This  is  preferable  to  the 
terms  front  and  back,  as  with  stationary 
engines  opinion  is  not  unanimous  as  to  which 
is  front  and  which  back;  thus  to  avoid  con- 
fusion, head  and  crank  end  will  be  used,  the 
ports  and  everything  dealing  with  the  steam 
entering  or  leaving  the  ports  at  either  end  of 
the  cylinder  being  designated  accordingly. 

The  ports  and  valves  controlling  the  in- 
coming steam  are  commonly  known  as  steam 
ports  and  valves,  while  for  the  outgoing  steam 
they  are  exhaust  ports  and  valves;  steam  and 
exhaust  in  general  refer  to  incoming  and  outgoing  steam  respectively. 


8  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Then  ports  1  and  2  are  the  steam  ports  for  the  head  and  crank  ends 
respectively,  while  3  and  4  are  the  exhaust  ports. 

The  piston  rod  passes  through  a  stuffing  box  in  the  cylinder  head  at  the 
crank  end,  which  prevents  leakage  of  steam  from  this  end  of  the  cylinder. 
One  or  more  packing  rings,  to  be  described  in  Chap.  XXIII,  keep  the 
steam  from  leaking  past  the  piston.  In  Fig.  4  the  piston  is  at  the  head 
end  of  the  stroke  and  the  crank  is  on  its  head-end  dead  center,  so  called 
because  the  steam  can  have  no  turning  action  on  it  when  in  this  position, 
and  the  engine  would  stop  if  it  were  not  for  the  momentum  of  the  fly  wheel, 
which  carries  the  crank  past  the  dead  center. 

When  the  crank  is  on  dead  center,  a  space  is  left  between  the  piston 
and  the  cylinder  head,  so  that  there  will  be  no  danger  of  striking  in  case 
of  wear  in  the  pin  joints  of  the  connecting  rod  and  the  main  bearing. 
The  distance  between  the  piston  and  cylinder  head  is  called  the  clearance 
distance  or  mechanical  clearance.  The  volume  of  this  space,  including 
the  volume  of  the  ports  up  to  the  valve  seats  is  called  the  clearance 
volume;  the  ratio  of  this  volume  at  one  end  of  the  cylinder,  to  the  volume 
swept  through  by  the  piston  during  one  stroke  is  commonly  called  the 
clearance,  and  is  usually  given  in  per  cent.  In  the  ordinary  steam  engine 
this  varies  from  3  to  12  per  cent.,  and  it  is  usually  desirable  to  keep  it  as 
small  as  possible;  however,  in  some  forms  of  steam  engine,  and  in  the  gas 
engine,  it  must  have  a  certain  definite  value,  which  in  some  cases  may  be 
much  greater  than  the  values  given. 

On  the  top  of  the  cylinder  is  the  steam  chest,  forming  a  passage  from 
the  steam-pipe  connection  to  the  steam  valves  at  each  end.  The  throttle 
valve  or  stop  valve  is  connected  to  the  steam-chest  flange.  The  valves 
are  of  the  rotary  type,  having  an  oscillating  motion.  The  exhaust  pas- 
sage at  the  bottom  of  the  cylinder  connects  the  exhaust  valve  at  each  end 
with  the  exhaust  pipe. 

Admission. — Figure  4  shows  the  piston  at  the  head  end  of  the  stroke, 
with  the  crank  on  dead  center.  The  head-end  steam  valve  is  open  by  a 
small  amount,  this  opening  being  called  the  lead.  This  means  that  the 
opening  of  the  valve,  or  admission,  occurred  a  little  before  the  crank 
reached  the  dead  center.  The  purpose  of  this  is  to  allow  full  steam  pres- 
sure in  the  clearance  space,  and  to  have  the  valve  partly  open,  before  the 
piston  starts  its  stroke,  in  order  that  there  may  be  no  delay  in  the  steam 
entering  the  cylinder  and  following  the  piston  without  loss  of  pressure. 
The  higher  the  rotative  speed  of  the  engine,  the  more  is  lead  necessary, 
high-speed  engines  having  proportionately  more  lead  than  low-speed 
engines. 

The  head-end  steam  valve  being  open,  steam  will  enter  and  force  the 


THE  STEAM  ENGINE 


9 


piston  toward  the  crank  end  of  the  cylinder.  It  will  also  be  seen  that  the 
crank-end  exhaust  valve  is  open,  allowing  steam  in  that  end  of  the  cylinder 
to  escape  to  the  exhaust  pipe. 

As  the  crank  rotates  in  a  clockwise  direction,  and  the  eccentric  is 
somewhat  more  than  90  deg.  in  advance  of  the  crank,  it  is  obvious  that  be- 
fore the  crank  reaches  an  upright  position,  the  wrist  plate  will  swing  to 
its  extreme  right  position,  giving  a  maximum  opening  to  the  head-end 
steam  valve  and  the  crank-end  exhaust  valve,  both  of  which  start  to 
close  as  the  crank  continues  to  rotate. 

Cut-off. — The  closing  of  the  steam  valve  is  known  as  cut-off,  and  this 
event  for  the  head  end  is  shown  in  Fig.  5.  This  is  the  first  event  since 
the  beginning  of  the  stroke  under  consideration.  The  crank-end  steam 
valve  and  head-end  exhaust  valve  have  remained  closed  so  far,  and  the 


FIG.  5. 

crank-end  exhaust  valve  is  still  open.  Cut-off  is  usually  designated  in 
the  fraction  of  the  stroke  accomplished  before  cut-off  occurs,  as  one-half 
cut-off,  one-third  cut-off,  etc. 

Early  steam  engines  received  steam  during  the  entire  stroke,  exhaust- 
ing it  to  the  atmosphere  when  at  its  maximum  pressure,  which  was 
wasteful.  The  cutting  off  of  steam  before  the  stroke  is  completed  was 
introduced  by  James  Watt  in  order  to  utilize  the  expansive  energy  of  the 
steam.  After  cut-off  the  steam  expands,  continuing  to  exert  pressure 
against  the  piston  until  the  end  of  the  stroke.  The  pressure,  however, 
decreases  as  expansion  continues,  resulting  in  a  decreased  mean  pressure 
during  the  stroke,  and  to  offset  this  a  larger  cylinder  must  be  used.  But 
the  larger  cylinder  with  cut-off  before  the  completion  of  the  stroke, 
always  uses  less  steam  for  a  given  horsepower  than  the  older  method. 
This  will  be  demonstrated  in  Chap.  XII.  The  earlier  the  cut-off  the 
greater  the  expansion,  and  if  carried  so  far  that  the  pressure  drops  to 


10 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


that  of  the  atmosphere  or  condenser  into  which  the  engine  exhausts,  the 
expansion  is  said  to  be  complete.  Theoretically,  this  would  secure  the 
maximum  economy,  but  for  practical  reasons  to  be  explained  in  Chap.  IX, 
the  best  economy  demands  a  cut-off  which  does  not  result  in  complete 
expansion. 

The  ratio  of  the  total  volume  of  steam  on  the  working  side  of  the  pis- 
ton at  the  completion  of  the  stroke,  to  the  volume  at  cut-off  when  expan- 
sion begins,  is  called  the  ratio  of  expansion.  For  the  purpose  of  avoid- 
ing a  more  advanced  discussion  of  valves  and  valve  gears  at  this  place,  the 
cut-off  shown  in  Fig.  5  is  late  in  the  stroke,  or  such  as  would  obtain  with  a 
heavy  load  on  the  engine. 

Brief  mention  of  the  indicator  diagram  may  be  of  advantage  at  this 
point.  The  steam-engine  indicator  is  an  instrument  for  determining  the 
steam  pressure  acting  upon  the  piston  during  its  stroke.  This  instru- 
ment traces  diagrams  similar  to  Fig.  6,  which  shows  conventional  dia- 


FIG.  6. 

grams  with  which  the  different  events  of  the  stroke  may  more  easily  be 
explained  than  with  actual  diagrams  taken  from  the  engine.  Horizontal 
measurements  represent  points  along  the  piston  travel,  while  vertical 
measurements  represent  pressure.  The  pressure  at  (a)  is  the  initial 
pressure,  at  (c)  the  terminal  pressure,  along  line  d-e  the  back  pressure  and  at 
(/)  the  compression  pressure.  In  common  parlance,  this  means  pressure 
measured  from  atmospheric  pressure  (14.7  Ib.  per  sq.  in.),  or  gage  pres- 
sure, but  it  is  more  satisfactory,  and  is  necessary  for  calculation  to  measure 
it  from  perfect  vacuum,  or  in  absolute  pressure. 

Line  o-v  represents  zero  pressure,  and  pressures  measured  from  it  are 
absolute  pressures.  Line  o-p  is  the  zero  volume  line,  and  all  volumes 
are  measured  from  it.  The  distance  p-a  represents  the  clearance  volume 
previously  mentioned.  This  space  is  full  of  steam  when  the  piston  starts 
its  stroke. 

In  practice  the  indicator  is  attached  at  (x),  Fig.  2,  in  the  clearance 
space  for  the  head-end  diagram,  so  that  this  diagram  gives  the  pressure 
acting  on  the  left  side  of  the  piston  during  the  two  strokes  forming  the 


THE  STEAM  ENGINE 


11 


cycle.  Likewise  the  indicator  is  attached  at  (y)  for  the  crank-end  dia- 
gram and  gives  the  pressure  at  the  right  of  the  piston.  It  will  be  assumed 
that  an  indicator  is  attached  at  each  end,  drawing  both  diagrams  at  the 
same  time,  which  is  practiced  in  high  class  tests. 

Thus  far  the  piston  has  moved  from  (a)  to  (6),  drawing  the  line  a-b, 
or  steam  line  of  the  head-end  diagram,  and  a  portion  of  line  d'-e',  or  back 
pressure  line  of  the  crank-end  diagram. 

Compression. — Continuing  the  rotation  of  the  crank  a  little  further  to 
the  position  of  Fig.  7,  causes  the  closure  of  the  crank-end  exhaust  valve. 
This  completes  the  back  pressure  line  d'-e'  of  the  crank-end  diagram,  and 
draws  a  portion  of  the  expansion  curve  b-c  of  the  head-end  diagram,  a 
curve  which  will  be  discussed  in  later  chapters.  The  closing  of  the  exhaust 


FIG.  7. 

valve  is  commonly  known  as  compression,  and  is  expressed  as  the  fraction 
of  the  stroke  completed  up  to  the  closure  of  the  exhaust  valve,  as  nine- 
tenths  compression. 

As  with  cut-off,  the  early  engine  closed  the  exhaust  valve  at  the  end 
of  the  stroke,  but  with  higher  speeds  it  was  found  that  an  engine  was  apt 
to  run  more  quietly  if  the  exhaust  valves  closed  before  the  stroke  was 
complete,  retaining  a  certain  amount  of  steam  and  compressing  it  in 
the  clearance  space  to  form  a  cushion  which  gradually  takes  up  the  lost 
motion  in  the  pin  joints.  The  necessity  of  compression  has  been  ques- 
tioned, so  any  further  remarks  along  this  line  are  left  until  Chap.  XVI. 
The  effect  of  compression  is  discussed  in  Chaps.  IX,  XII  and  XVI. 

Release. — After  compression,  all  valves  are  closed.  A  further  rotation 
of  the  crank  starts  the  head-end  exhaust  valve  open,  which  is  shown 
in  Fig.  8;  this  is  known  as  release.  This  completes  the  expansion  line 
b-c  of  the  head-end  diagram  and  draws  a  portion  of  the  crank-end  com- 
pression curve  e'-f  of  the  crank-end  diagram.  Release  is  usually  earlier 


12 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


in  the  stroke  than  compression  but  nearly  always  occurs  before  the  stroke 
is  complete  in  order  that  the  steam  may  leave  the  cylinder  sufficiently 
to  have  its  pressure  lowered  to  that  of  the  atmosphere  or  condenser  be- 
fore the  piston  starts  the  next  stroke;  otherwise  the  back  pressure  is 


FIG.  8. 

high  at  the  beginning  of  the  stroke,  reducing  the  effective  work  on  the 
piston.     As  with  lead,  release  must  be  earlier  with  higher  speeds. 

A  little  further  rotation  of  the  crank  opens  the  crank-end  steam 
valve.  This  completes  crank-end  compression  curve  e'-f  and  draws  a 
portion  of  head-end  exhaust  line  c-d.  The  stroke  is  completed  in  Fig.  9, 


FIG.  9. 

in  which  the  crank  is  on  the  crank-end  dead  center.  This  completes  the 
head-end  exhaust  line  c-d  and  draws  the  crank-end  admission  line  /'-a'. 
These  two  lines  are  shown  as  straight  lines  in  Fig.  6,  but  they  vary  in 
form  in  actual  diagrams  and  the  junctions  of  the  different  lines  are  not 
sharply  defined. 


THE  STEAM  ENGINE  13 

Completing  the  revolution  to  head-end  dead  center  produces  con- 
secutively, crank-end  cut-off,  head-end  compression,  crank-end  release 
and  head-end  admission,  completing  the  cycle,  and  drawing  the  line 
a'b'c'd'  of  the  crank-end  diagram  and  defa  of  the  head-end  diagram. 

The  principle  of  operation  is  practically  the  same  in  all  steam  engines, 
except  that  some  accomplish  all  four  events  in  both  ends  of  the  cylinder 
with  a  single  valve;  exception  may  also  be  made  in  the  case  where  the 
piston  has  the  function  of  a  valve,  as  in  the  uniflow  engine. 

3.  Types  and  Classification. — The  development  of  the  steam  engine, 
extending  as  it  has  over  a  comparatively  long  period,  has  resulted  in  a 
large  variety  of  designs,  and,  eliminating  all  but  those  at  present  being 
placed  upon  the  market  by  recognized  builders  still  leaves  a  goodly 
number.  But  little  attempt  has  been  made  at  general  standardization, 
although  there  is  considerable  similarity  in  a  given  type  as  manufactured 
by  different  builders,  due  no  doubt  to  a  sort  of  mutual  influence,  those 
having  the  most  originality  and  initiative  leading  the  way. 

A  logical  classification  is  difficult,  as  the  engine  may  belong  to  several 
classes.  With  this  in  mind,  terms  in  common  use  which  express  some 
feature  of  design  will  be  arranged  under  different  headings,  and  a  few 
of  the  least  obvious  briefly  described. 

• 

Classification  According  to  General  Form. 
Horizontal  or  vertical. 
Single-acting  or  double-acting. 
Side-crank  or  center-crank. 
Right-hand  or  left-hand. 
Run  over  or  run  under. 
Belt  forward  or  belt  back. 

Single-acting  or  Double-acting. — A  single-acting  engine  uses  but  one 
end  of  the  cylinder,  the  head  end,  while  the  crank  end  is  open.  This  ob- 
viates the  necessity  of  a  crank-end  cylinder  head  and  a  stuffing  box, 
consequently  the  crosshead  may  be  eliminated  and  its  function  performed 
by  the  piston,  which  is  made  unusually  long  to  provide  ample  wearing 
surface.  Crossheads  are  sometimes  used  on  single-acting  engines. 

Single-acting  steam  engines  usually  have  two  cylinders,  one  of  them 
being  a  low-pressure  cylinder,  forming  a  compound  engine,  in  some  cases. 
The  cranks  are  opposite,  or  180  degrees  apart.  With  steam  engines, 
this  type  is  usually  vertical  and  its  advantage  is  reduced  head  room. 

Side-crank  or  Center-crank. — In  this  country,  most  steam  engines  of 
large  and  medium  size  are  of  the  side-crank  class.  This  is  shown  in 
diagram  in  Fig.  10.  The  shaft  has  but  one  bearing  located  in  the  engine 
frame,  the  other,  called  the  outer  bearing  or  outboard  bearing,  being  a 


14 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


separate  bearing,  usually  having  no  connection  to  the  engine  frame,  but 
bolted  separately  to  the  foundation. 

Some  small  engines  are  also  built  with  the  side  crank,  and  of  course, 
some  large  engines  have  the  center  crank.  Figure  11  shows  a  center-crank 
engine  in  diagram.  Sometimes  the  outer  bearing  is  omitted,  a  belt 


FIG.  10. 


FIG.  11. 


RIGHT  HAND  ENGINE. 


wheel  being  overhung  from  the  main  bearing  on  one  side,  while  the  gover- 
nor wheel  is  on  the  other. 

The  advantage  claimed  for  the  center-crank  engine  is  that  of  a  beam 
supported  at  the  ends  with  a  load  in  the  middle,  over  a  cantilever  beam 
loaded  at  the  end;  but  where  the  outer  bearing  is  used,  as  it  always  is 

in  large  engines,  the  difficulty  and 
uncertainty  of  keeping  three  bear- 
ings in  perfect  alinement  makes 
the  advantage  dubious,  and  as 
there  is  no  difficulty  in  the  prac- 
tical design  of  the  side-crank  type, 
it  is  used  almost  entirely  in  this 
country,  not  only  for  steam  en- 
gines, but  for  large  gas  engines 

Right-hand  or  Left-hand. — This 
classification  has  not  a  universal 
application  except  in  the  case  of 
horizontal  side-crank  engines. 
With  these,  standing  at  the  cyl- 
inder end  and  looking  toward  the 
crank,  the  engine  is  right-hand  if  the  main  bearing  is  at  the  right  of  the 
center  line  of  engine,  and  left-hand  if  the  bearing  is  at  the  left.  This  is 
illustrated  in  Fig.  12. 

Center-crank  engines  may  be  classified  as  right-  or  left-hand  according 
to  the  valve-gear  or  governor  side,  and  vertical  engines  are  so  classified 


H^^SF 


LEFT  HAND  ENGINE 


FIG.  12. 


THE  XTEAM  ENGINE 


15 


for  convenience  in  ordering;  in  such  cases  directions  are  usually  given  in 
the  builders'  catalogs. 

Run  Over  or  Under. — This  is  a  functional  classification  with  most 
engines,  it  being  possible  to  run  them  either  over  or  under  by  adjusting 
the  valve  gear.  This  classi- 
fication is  a  definite  one  only 
with  horizontal  engines,  in 
which  case,  if  the  top  of  the 
wheel  moves  from  an  observer 
standing  at  the  cylinder,  the 
engine  runs  over;  if  it  comes 
toward  him  it  runs  under. 
This  is  illustrated  by  Fig.  13. 

As    with   the   designation 
right-  and  left-hand,  the  terms 
run  over  and  under  are  sometimes  applied  to  vertical  engines  by  special 
direction  in  the  builders'  catalogs. 

It  is  true  that  some  engines,  as  the  marine  type,  are  built  to  run  in 
one  direction,  the  bearing  surface  of  the  crosshead  shoe  being  much 
greater  on  one  side  than  the  other.  The  smaller  surface  comes  into  play 


FIG.  13. 


FIG.  14. 

only  while  the  engine  is  reversing,  which  is  a  very  small  portion  of  the 
time. 

Belt  Forward  or  Back. — While  this  might  appear  to  be  entirely  func- 
tional, it  often  affects  the  location  of  the  wheel,  and  therefore  the  length 


16  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

of  the  shaft,  which  in  turn  sometimes  affects  the  necessary  diameter. 
In  Fig.  14,  A  shows  the  belt  going  forward,  and  B  going  back.  In  the 
latter,  space  must  be  left  between  the  belt  and  the  engine  cylinder  so 
that  the  operating  engineer  may  get  at  the  valve  gear  and  throttle  valve. 

Classification  According  to  Valve  Gear. 
Single-valve. 
Four-valve. 
Gridiron-valve. 
Poppet-valve. 
Throttling. 
Automatic  cut-off. 
Reversing. 

Throttling  Engine. — All  stationary  engines  are  controlled  by  govern- 
ors, whieh  limit  the  speed  fluctuation  to  a  small  fraction  under  changing 
loads,  within  the  capacity  of  the  engine.  The  governor  and  valve  gear 
are  treated  in  later  chapters,  but  their  effect  upon  the  steam  supply 
may  be  considered  here.  The  throttling  engine  governs  in  the  simplest 
way  and  requires  the  least  mechanism.  The  governor  is  attached  to  a 
throttle  valve  in  the  steam  pipe.  If  the  load  on  the  engine  decreases, 
the  engine  tends  to  speed  up,  and  in  doing  so  the  governor  weights  fly 
out  by  centrifugal  force,  and  being  connected  to  the  controlling  valve  in 
the  steam  pipe,  partly  closes  it,  throttling  the  steam,  which  cannot  get 
through  the  valve  fast  enough  to  keep  up  pressure  as  it  follows  the  piston, 
therefore  less  work  is  done,  the  steam  pressure  being  adapted  to  the  load 
by  the  governor.  If  the  load  is  increased,  the  engine  tends  to  slow 
down,  and  this  opens  the  valve,  increasing  the  steam  pressure. 

The  throttling  engine  is  usually  a 
single-valve  engine  with  either  a  flat 
"D  "-valve  or  a  piston  valve.  The 
eccentric  is  fixed  to  the  shaft  and  the 
valve  always  has  the  same  travel; 
therefore  the  cut-off,  compression,  etc. 
are  always  the  same  and  the  con- 
~  ~~~~  trolling  device  is  really  not  a  part  of 

the  engine  proper. 

The  effect  of  the  governor  upon  the  steam  pressure  within  the  cylinder 
may  be  shown  by  an  indicator  diagram,  Fig.  15,  in  which  the  full  lines 
represent  the  normal  load  on  the  engine,  and  the  dotted  lines  a  greater 
and  a  lesser  load.  During  the  time  the  steam  is  being  driven  from  the 
cylinder,  it  is  open  to  the  atmosphere,  so  the  back  pressure  line  and  com- 
pression curve  are  the  same  for  all  loads. 


THE  STEAM  ENGINE 


17 


Throttling  engines  are  usually  small  engines  and  their  economy  is 
comparatively  poor.  This  is  attributable  in  part,  however,  to  lack  of 
refinement  in  other  features  of  design  common  in  a  good  many  small 
engines.  Within  reasonable  limits  of  load  variation  this  method  of  gov- 
erning possesses  some  practical  advantages.  An  economical  cut-off  may 
be  maintained  at  all  loads,  and  for  loads  less  than  the  maximum  the 
throttling  of  the  steam  tends  to  reduce  the  amount  of  moisture  due  to 
partial  condensation,  which  will  be  explained  in  Chap.  IX.  Governing  by 
throttling  is  further  discussed  in  Chap.  XII,  Par.  58. 

Automatic  Cut-off  Engine. — These  engines  are  controlled  by  changing 
the  cut-off,  allowing  smaller  or  greater  quantities  of  steam  of  constant 
pressure  to  enter  the  cylinder  to  meet  the  load  requirements.  The 
Corliss  engine  was  one  of  the  earliest  forms  of  this  class,  and  one  of  the 
most  important  for  medium  and  large  powers.  The  gear  of  this  engine  is 


FIG.  16. 


FIG.  17. 


known  as  a  releasing  gear,  and  sometimes  as  a  drop-cut-off  gear.  There 
are  four  valves  as  previously  stated,  two  steam  and  two  exhaust  valves, 
which  receive  their  motion  directly  from  a  wrist  plate  as  explained  in  Par. 
2.  The  eccentric  is  fixed  to  the  engine  shaft,  giving  a  constant  movement 
to  the  wrist-plate  and  valve  levers  connected  thereto  by  the  links.  The 
movement  of  the  exhaust  valves  is  constant,  and  the  exhaust  events, 
compression  and  release,  are  always  the  same. 

This  gear  is  described  in  Chap.  XX. 

The  releasing  principle  has  also  been  applied  to  four-valve  engines 
with  gridiron  steam  valves,  and  also  to  poppet-valve  engines. 

The  indicator  diagram  for  the  engine  just  described  is  shown  in  Fig.  16, 
the  full  lines  being  for  rated  load  and  the  dotted  lines  for  a  greater  and 
smaller  load. 

Another  method  of  regulating  the  steam  supply  by  change  of  cut-off  is 
that  of  changing  the  position  of  the  eccentric  upon  the  shaft,  which  is 
effected  by  a  shaft  governor  to  be  described  in  Chap.  XIX.  The  diameter 
of  the  circle  upon  which  the  eccentric  center  travels  is  usually  changed,  but 

not  in  all  engines,  and  the  angular  position  of  the  eccentric  relative  to 
2 


18  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

the  crank  is  always  changed  upon  change  of  load.  This  method  is  applied 
to  engines  fitted  with  all  the  variety  of  valves  previously  mentioned. 
When  the  single  valve  is  used,  or  a  four-valve  engine  with  a  single  eccen- 
tric such  as  was  employed  in  describing  steam-engine  operation  in  Par.  1, 
the  change  in  eccentric  position  necessarily  affects  all  events  of  the  stroke 
as  shown  in  Fig.  17,  the  full  lines  being  for  rated  load  as  before,  and  the 
dotted  lines  for  greater  and  lesser  loads.  Two  styles  of  dotted  lines 
are  used  to  associate  the  expansion  curve  with  its  corresponding  compres- 
sion curve.  It  will  be  noticed  that  shortening  the  cut-off  is  accompanied 
by  an  earlier  compression,  reducing  the  diagram  area  on  two  sides,  while 
for  long  cut-off,  the  area  is  increased  on  two  sides.  This  necessitates  less 

change  of  the  gear  for  a  given  change 
of  load  than  if  the  compression  were 
constant.  For  engines  having  ex- 
haust valves  operated  by  a  sep- 
arate eccentric,  the  compression  and 
release  are  constant,  and  the  diagram 
is  that  of  Fig.  16. 

A  combination  of  throttling  and 
automatic   cut-off    is  employed    in 
^FIG   is.x  some   engines  with  good  economy, 

especially  at  light  loads,  when  the 

reduction  of  pressure  due  to  throttling,  with  its  consequent  drying  effect, 
makes  unnecessary  quite  so  short  a  cut-off  as  would  be  required  if  only 
the  cut-off  were  altered.  This  is  shown  by  the  indicator  diagram  of  Fig. 
18,  in  which  full  lines  are  for  normal  or  rated  load  and  dotted  lines 
for  a  lighter  load. 

Classification  According  to  the  Use  of  Steam. 
Simple. 

Compound  and  multiple-expansion. 
Condensing. 
Noncondensing. 

Simple. — A  simple  engine  receives  steam  at  or  near  boiler  pressure, 
which,  after  doing  work  in  the  engine  cylinder,  is  exhausted  to  the  atmos- 
phere, a  condenser,  into  an  exhaust  heating  system  or  utilized  for  some 
industrial  process.  A  simple  engine  may  have  more  than  one  cylinder, 
such  as  a  duplex  or  twin  engine,  but  each  receives  steam  from  the  same 
source  at  the  same  pressure;  at  any  rate,  no  cylinder  receives  steam  from 
the  other.  Most  modern  simple  engines  have  a  single  cylinder. 

Compound  and  Multiple-expansion. — It  has  been  already  stated  in 


THE  STEAM  ENGINE  19 

Par.  2  that  the  use  of  the  expansive  energy  of  steam  is  conducive  to 
economy,  but  that  too  great  an  expansion  in  one  cylinder,  or  too  early  a 
cut-off  is  not  economical,  so  that  the  ratio  of  expansion  is  necessarily 
limited  in  a  single  cylinder.  If  a  cut-off  within  practical  economical 
limits  is  effected  in  one  cylinder,  and  the  steam  expanded  to  some  pressure 
between  that  of  the  steam  and  exhaust  mains,  and  if  the  exhaust  from 
this  cylinder  is  piped  to  a  cylinder  of  greater  volume,  having  a  cut-off 
also  within  economical  limits,  and  such  that  the  volume  of  the  cylinder 
up  to  cut-off  is  approximately  equal  to  the  volume  of  the  first  cylinder, 
the  steam  would  expand  again  from  this  intermediate  pressure,  which  is 
the  initial  pressure  for  the  second  cylinder,  to  some  terminal  pressure  some- 
what higher  than  the  exhaust-main  pressure,  the  latter  being  the  back 
pressure  of  the  second  cylinder.  Then  the  total  ratio  of  expansion,  from 
the  volume  at  cut-off  in  the  first  cylinder  to  the  final  volume  in  the  second, 
will  be  approximately  the  product  of  the  ratios  of  expansion  in  the  two 
cylinders.  The  exhaust  from  the  second  cylinder  is  usually  piped  to  a 
condenser,  but  sometimes  to  the  atmosphere,  or  the  steam  may  be  used 
for  some  purpose  as  with  the  simple  engine. 

Thus,  by  the  use  of  two  cylinders  of  different  size,  but  so  connected 
as  to  deliver  power  to  the  same  shaft,  economical  cut-off  may  be  com- 
bined with  a  high  expansion  ratio,  greatly  increasing  the  economy. 
The  first  cylinder  is  known  as  the  high-pressure  cylinder  and  the  second 
as  the  low-pressure  cylinder.  As  the  piston  stroke  is  the  same  in  both 
cylinders,  the  volumes  of  the  cylinders  are  proportional  to  the  piston 
areas  acted  upon  by  the  steam.  The  mean  pressures  acting  on  the  two 
pistons  are  about  proportional  to  the  piston  areas  inversely,  therefore  the 
work  done  in  the  two  cylinders  is  nearly  equal.  Such  an  engine  is  called 
a  compound  engine.  Although  in  a  general  sense  this  term  covers  engines 
in  which  expansion  takes  place  in  three  or  four  cylinders,  it  is  usually 
employed  for  a  double-expansion  engine  as  just  described,  and  when  ex- 
pansion occurs  in  three  or  four  stages,  the  engines  are  known  as  triple- 
expansion  engines  or  quadruple-expansion  engines. 

The  exhaust  from  the  high-pressure  cylinder  is  piped  to  a  vessel  called 
a  receiver,  from  which  the  steam  supply  for  the  low-pressure  cylinder  is 
taken.  The  influence  of  the  receiver  is  to  prevent  excessive  fluctua- 
tion of  pressure  during  the  passage  of  steam  from  the  high-pressure 
to  the  low-pressure  cylinder.  In  triple-expansion  engines  the  first 
cylinder  is  the  high-pressure,  the  second  the  intermediate  and  the  third 
the  low-pressure  cylinder.  There  is  a  receiver  between  the  high  and 
intermediate,  and  another  between  the  intermediate  and  low-pressure 
cylinders.  In  quadruple-expansion  engines,  the  second  cylinder  is  the 


20  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

first  intermediate  and  the  third,  the  second  intermediate.  A  receiver  is 
placed  between  each  cylinder  as  before,  there  being  three  receivers  with  a 
quadruple-expansion  engine. 

In  marine  engines,  in  order  to  secure  better  balancing  and  a  more  uni- 
form turning  effort  on  the  crank  shaft,  the  low-pressure  cylinders  of  com- 
pound and  triple-expansion  engines  are  sometimes  replaced  by  two 
cylinders  with  a  combined  capacity  of  the  single  cylinder.  These  two 
cylinders  take  steam  from  the  same  receiver  and  exhaust  to  a  common 
condenser,  and  therefore  belong  to  the  same  stage.  Such  engines  are 
known  respectively  as  three-cylinder  compounds  and  four-cylinder 
triple-expansion  engines. 

Multiple-expansion  engines  other  than  double-expansion  or  com- 
pounds are  used  comparatively  little  except  in  marine  and  pumping  serv- 
ice. There  is  little  gain  over  a  compound  in  economy  when  the  ratio 
of  cylinder  volumes  is  properly  chosen,  and  the  use  of  superheated  steam 
has  still  further  reduced  such  gain. 

The  most  common  types  of  compound  engines  are  the  cross-compound 
and  the  tandem-compound.  The  cross-compound  consists  of  two  engines, 
one  high-pressure  and  one  low-pressure,  parallel  to  each  other  and  con- 
nected to  the  same  shaft.  To  provide  a  more  uniform  turning  effort  on 
the  shaft,  the  cranks  are  placed  90  degrees  apart,  although  this  i»  not 
necessarily  the  best  angle  to  produce  this  result.  This  subject  is  dis- 
cussed in  Chap.  XVIII. 

A  plan  drawing  of  a  cross-compound  engine  with  its  receiver  and 
piping  is  shown  in  Fig.  461,  Chap.  XXXIV. 

A  tandem-compound  engine  has  its  cylinders  in  line,  the  pistons  being 
fastened  to  a  common  piston  rod.  .  The  cylinders  are  connected  to  a  single 
engine  frame,  and  all  other  parts  are  as  for  a  single  engine.  There  is  a 
single  crank  and  the  turning  effort  is  not  as  uniform  as  with  a  cross- 
compound  engine.  A  tandem  engine  is  more  compact  and  is  usually 
considered  to  be  better  adapted  to  high  speed. 

Noncondensing  and  Condensing. — Any  steam  engine  exhausting  into 
the  atmosphere,  or  any  higher  pressure,  such  as  a  heating  system,  is 
known  as  a  noncondensing  engine.  If  the  exhaust  pressure  is  below  that 
of  the  atmosphere,  a  condenser  must  be  used  and  the  engine  is  known  as  a 
condensing  engine.  This  classification  affects  the  design  of  the  simple 
engine  but  little;  in  the  compound,  however,  the  size  of  the  low-pressure 
cylinder,  and  the  ratio  of  cylinder  diameters  are  both  greater  for  the 
condensing  engine,  but  it  is  not  uncommon  for  a  condensing  engine  to  run 
noncondensing  for  a  portion  of  the  time. 

Condensing  engines   show  a  greater  economy  than  noncondensing 


THE  STEAM  ENGINE 


21 


engines,  due  to  the  removal  of  back  pressure,  and  to  the  greater  ratio 
of  expansion  permitted,  especially  in  compound  engines. 

Classification  According  to  Service  for  which  Designed. — Among  these 
are  the  following: 

Locomotive. 

Marine  engine. 

Hoisting  engine. 

Pumping  engine. 

Rolling-mill  engine. 

Others  might  be  added  to  the  list. 

4.  The  Uniflow  Engine. — This  engine  was  designed  by  Prof.  Stumpf, 
and  is  manufactured  by  several  firms  in  the  United  States.     It  has  no 


FIG.   19. — Nordberg  uniflow  cylinder  with  Corliss  valves. 

exhaust  valves,  the  piston  performing  this  function  near  the  end  of  the 
stroke.  The  steam  thus  flows  in  but  one  direction  at  each  end  of  the  cylin- 
der, the  period  of  exhaust  being  greatly  decreased,  thus  lessening  the 
cooling  effect  of  the  low-pressure  steam  upon  the  cylinder  walls  and  reduc- 
ing condensation. 


FIG.  20. — Indicator  diagrams  for  Universal  uniflow  engine. 

A  section  of  the  cylinder  of  the  Nordberg  uniflow  engine  is  shown  in 
Fig.  19.  The  valves  D  act  as  automatic  relief  valves  when  the  vacuum  is 
accidentally  broken,  increasing  the  clearance  space  by  the  volume  of 


22 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


space  B.  Otherwise,  as  compression  begins  at  about  Ho  stroke,  the 
compression  pressure  would  be  much  above  the  initial  pressure. 
These  valves  are  backed  off  by  the  hand  wheels  when  it  is  desired  to 
run  the  engine  noncondensing 

Uniflow  engines  are  mostly  made  with  poppet  valves. 


to 


10 


si  * 

Load 
FIG.  21. — Steam  consumption  of  Universal  uniflow  engine. 

Indicator  diagrams  for  condensing  and  noncondensing  operation  for  a 
Universal  Uniflow  engine  are  shown  in  Fig.  20.  This  engine  has  an  auxil- 
iary exhaust  valve  which  closes  about  0.7  stroke,  and  is  in  operation  when 
the  engine  works  noncondensing.  Fig.  21  shows  the  steam  consumption 
of  the  Universal  Uniflow  engine. 


CHAPTER  IV 
THE  STEAM  TURBINE 

5.  According  to  historical  records,  the  first  self-acting  machine  oper- 
ated by  steam  was  a  turbine,  but  it  was  not  until  a  comparatively  recent 
date  that  the  development  of  the  steam  turbine  into  a  commercially 
successful  machine  was  undertaken.     Naturally,  under  the  present  state 
of  the  mechanical  arts,  its  advance  was  much  more  rapid  than  that  of 
the  steam  engine,  until  now  it  equals  in  economy  and  far  exceeds  in  unit 
capacity,  the  best  steam  engines. 

The  expansion  of  steam  in  the  cylinder  of  a  reciprocating  steam  engine, 
while  essential  to  economy,  is  not  a  necessity  to  practical  operation; 
this  is  obvious  from  certain  types  of  steam  pumps  which  receive  steam  at 
boiler  pressure  throughout  the  stroke,  and  even  from  high-grade  engines 
successfully  carrying  overloads  which  require  steam  admission  nearly  the 
entire  stroke  with  very  little  expansion  in  the  cylinder.  But  the  tur- 
bine depends  upon  the  velocity  of  the  steam,  which  may  only  be  effected 
by  expanding  it  through  properly  formed  nozzles  from  a  higher  to  a  lower 
pressure,  the  form  of  the  nozzles  depending  upon  the  lower  pressure 
against  which  they  discharge.  With  nozzles  properly  designed  and  con- 
structed, the  steam  issues  in  well-formed  jets  at  high  velocity,  which 
impinge  upon  blades  or  vanes  on  the  turbine  wheel,  causing  it  to  turn  and 
do  useful  work. 

The  operation  is  therefore  dependent  upon  two  principles,  the  conver- 
tion  of  heat  into  kinetic  energy  being  thermal  and  the  utilization  of  kinetic 
energy  for  work  on  the  turbine  wheel,  mechanical. 

6.  Impulse  and  Reaction. — Assume  three  fixed  vanes  of  different  form 
as  in  Fig.  22,  and  that  jets  of  steam  or  other  fluids  of  velocity  V  enter 
and  leave  the  vanes  as  indicated  by  the  arrows.     Further  assume  that 
the  .velocity  V,  the  angle  a  and  the  weight  of  fluid  flowing  in  a  given 
time  is  the  same  in  each  case,  and  that  the  flow  is  frictionless. 

The  force  PA  of  Fig.  22-A,  required  to  prevent  the  horizontal  displace- 
ment of  the  vane  to  the  right  is  said  to  be  due  to  the  impulse  of  the  jet. 
The  force  PB  of  Fig.  22-B  is  due  to  the  reaction  of  the  jet  and  is  equal  to 
PA.  The  force  Pc  in  Fig.  22-C  is  due  to  both  impulse  and  reaction  and 
is  equal  to  the  sum  of  PA  and  PB.  The  general  principle  is  the  same 

23 


24  DESIGN  AND  CONSTRUCTION  OF  PI  EAT  ENGINES 

whether  the  vanes  are  fixed  or  moving,  and  whether  the  velocity  of  the 
jet  relative  to  the  surface  of  the  vane  is  constant  or  variable. 

In  most  turbines  there  are  more  than  one  set  of  moving  vanes  and  at 
least  one  set  of  fixed  vanes.  The  fixed  vanes  serve  the  purpose  of  nozzles 
as  far  as  directing  the  steam  is  concerned,  and  will  be  so  designated  when 
they  compose  openings  through  a  diagram  dividing  the  turbine  into  com- 
partments. When  they  are  attached  to  the  turbine  casing  and  are  of  a 
form  similar  to  the  moving  vanes  they  are  called  guides,  and  all  moving 
vanes  are  called  blades. 


7.  Commercial  Classification. — Although  all  practical  steam  turbines 
utilize  both  impulse  and  Reaction,  the  greater  use  of  one  principle  or  the 
other  has  led  to  the  general  classification  of  impulse  turbines  and  reaction 
turbines.  All  turbines  in  which  expansion  occurs  only  in  the  nozzles  are 
called  impulse  turbines,  while  in  reaction  turbines,  expansion  also  occurs 
in  the  blade  passages. 

Considering  only  designs  largely  used,  impulse  turbines  may  be  further 
classified  as  follows: 

Simple 


Impulse 


Velocity-stage. 
Compound 

Pressure-stage. 


Reaction  turbines  are  always  compound  or  multi-stage. 
8.  Simple  Impulse  Turbines. — In  these  turbines  the  steam  expands 
in  the  nozzles  from  steam-chest  pressure  to  the 
exhaust  pressure,  which  is  atmospheric  for  non- 
condensing  and  vacuum  for  condensing  turbines. 
If   a  turbine   blade   of   semicircular  section, 
moving  in  the  direction  of  the  arrow  in  Fig.  23 
F      23  receives  a  steam  jet  flowing  in  the  same  direction 

of  velocity   V,  it  is  obvious  that  if  the  velocity 

of  the  blade  were  7/2,  the  relative  velocity  of  steam  to  blade  would  be 
F/2;  then  the  steam,  upon  leaving  the  blade  would  have  an  absolute 
velocity  of  zero;  that  is,  all  of  its  kinetic  energy  would  be  absorbed  by 


THE  STEAM  TURBINE 


25 


the  blade.  While  it  is  necessary  in  practice  that  the  steam  jet  be  de- 
livered to  the  wheel  at  an  angle,  and  there  must  always  be  some  absolute 
velocity  at  exit  or  residual  velocity,  it  is  clear  that  to  obtain  the  max- 
imum work  from  the  jet,  the  blade  velocity  must  approximate  one-half 
the  jet  velocity.  This  means  a  rim  velocity  of  nearly  1400  ft.  per  sec., 
necessitating  a  wheel  of  high-grade  material  and  special  design,  which 
will  be  considered  in  Chap.  XXXI. 

To  illustrate  the  change  in  pressure  and  velocity  of  the  steam  in  its 
passage  through  nozzles  and  blades,  the  diagrams  of  Fig.  24  are  given. 
Velocity  and  pressure  changes  are  represented  by 
straight  lines  for  convenience  and  for  lack  of  exact 
information. 

The  only  pressure  change  is  in  the  nozzle  and  this 
is  accompanied  by  an  increase  in  velocity,  a  maximum 
being  reached  at  exit.  There  is  no  pressure  change 
in  passage  through  the  blades,  but  a  decrease  in  abso- 
lute velocity  as  the  kinetic  energy  of  the  jet  is  given 
up  to  the  wheel.  The  relative  velocity  of  jet  and 
blade  is  assumed  to  be  constant,  friction  being  ne- 
glected. The  residual  velocity  (which  is  absolute), 
as  the  steam  leaves  the  blades,  is  partly  dissipated 
by  friction  as  it  forms  eddies  in  the  steam  remaining 
in  the  casing,  and  helps  to  clear  the  casing  of  steam  suffi- 
ciently to  maintain  a  constant  pressure. 

9.  Compound  Impulse  Turbines.  Velocity-stage. 
— It  was  shown  in  Par.  8  that  a  very  high  rim  ve- 
locity is  necessary  for  good  economy  in  a  simple  im- 
pulse turbine.  This  necessitates  a  rotative  speed  so  high  that  for  most 
turbine  applications,  gear  transmission  becomes  necessary.  To  obviate 
this,  or  to  make  possible  direct-connection,  especially  in  large  powers 
compounding  is  resorted  to. 

If  two  or  more  wheels  carrying  blades  are  placed  upon  a  shaft,  and 
between  each  row  of  blades,  fixed  guides  are  located  to  direct  the  steam 
leaving  one  wheel  into  the  blades  of  the  following  wheel,  the  velocity  of 
the  jet  is  only  in  part  taken  up  by  the  first  wheel,  part  by  the  second  wheel 
and  so  on,  until  upon  emerging  from  the  last  wheel,  the  residual  velocity 
may  be  the  same  as  that  of  a  simple  impulse  turbine.  Then  the  rim 
speed  is  proportional  to  the  number  of  wheels,  inversely,  which* is  also 
the  number  of  stages.  Thus,  if  a  simple  turbine  runs  12,000  r.p.m.,  a 
3-velocity-stage  turbine  with  the  same  sized  wheels,  and  the  same  steam 
and  exhaust  pressures  will  run  4000  r.p.m.  Due  to  greater  difference 


FIG.  24. — Simple  im- 
pulse turbine. 


26 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


between  jet  and  rim  velocity,  and  a  greater  number  of  blades  receiving 
the  impulse  of  the  steam,  the  total  turning  force  is  greater,  but  it  acts  at 
less  velocity;  the  power,  which  is  proportional  to  the  product  of  force 
and  velocity,  is  the  same  as  for  the  single  wheel  running  at  the  higher 
speed,  neglecting  differences  in  frictional  resistance  and  other  losses. 
A  diagram  of  a  2-stage  turbine  is  shown  in  Fig.  25. 
As  with  the  simple  turbine,  the  only  pressure  drop  is  in  the  nozzles, 
accompanied  by  velocity  increase.      There  is  partial  velocity  decrease  in 
each  wheel,  with  no  change  of  velocity  through  the  guides. 

It  must  be  remembered  that  these  velocity 
changes  in  the  moving  blades  are  absolute,  the 
velocity  relative  to  the  blade  surface  being  as- 
sumed constant.  Modifications  are  made  in 
practice  to  allow  for  friction,  and  different  forms 
of  blade  and  guide  sections  are  sometimes  used. 
These  will  be  considered  in  Chap.  XV. 

Two    and    three    stages   are  most  used   in 
velocity-stage  turbines,   although  four  and  five 
stages    are    sometimes  seen.     The  nozzles   and 
guides  extend  around  but  a  portion  of  the  cir- 
cumference of  the  casing,  acting  upon  but  a  part 
of  each  wheel  at  one  time.     A  set  of  nozzles  or 
Ve/         guides,   and  the  row  of  blades  into  which  they 
*  discharge  comprise  a  stage  in  all  types  of  com- 

_L\*  L  j  L  i !    !  Abs.  Pres. .     pound  turbines. 

FIG. 25— Velocity-stage im-          10.  Compound  Impulse  Turbines.    Pressure- 
stage.— -When    steam    expands    from    boiler    to 

exhaust  pressure,  it  gives  up  a  certain  amount  of  heat  which  may  be 
transformed  into  mechanical  energy  E.  If  this  is  utilized  to  produce 
velocity  V  in  steam  jets  for  turbine  propulsion,  we  know  by  mechanics 
that  the  kinetic  energy  of  W  Ib.  of  steam  equivalent  to  E  is  given  by 


E  = 


WVZ 


If  this  energy  is  divided  into  n  equal  parts,  each  of  which  is  used  in 
a  compartment  containing  a  turbine  wheel  by  being  consecutively  ex- 
hausted into  the  next  lower  compartment  until  the  pressure  in  the  last 
equals  the  exhaust  pressure,  the  equation  for  the  energy  delivered  to  each 
compartment  or  stage  will  be, 

#       WV2 
n         20 


THE  STEAM  TURBINE 


27 


The  velocity  of  the  jets  will  be, 

J2g       \E  _  constant 

=  W  V  n  =       Vn 


The  velocity  is  therefore  proportional  inversely  to  the  square  root  of 
the  number  of  stages.  Then,  assuming  the  same  ratio  of  jet  to  rim  veloc- 
ity in  all  cases,  the  rim  velocity  of  the  pressure-stage  turbine  with  wheels 
of  equal  size  may  be  proportional  to  the  square  root  of  the  number  of 
stages,  inversely.  Thus,  if  a  simple  turbine  runs  12,000  r.p.m.,  a  16-stage 
turbine  will  run 

12,000 

. =  3000  r.p.m. 


Figure  26  contains  diagrams  of  a  pressure-stage  turbine  with  two 
stages. 

Each  stage  is  like  the  single  stage  of  the  simple  turbine,  but  with 
less  pressure  and  velocity  change. 

In  order  that  the  radial  length  of  the 
blades  may  not  be  too  small,  the  nozzles  of  the 
first  stage  occupy  but  a  portion  of  the  circum- 
ference; the  volume  has  increased  sufficiently 
after  the  first  two  or  three  stages,  so  that  the 
entire  circumference  is  needed,  the  blades  and 
nozzles  increasing  gradually  in  radial  length 
to  accommodate  the  increasing  volume  of 
steam.  As  the  total  steam  passage  is  propor- 
tional to  the  product  of  the  radial  length  of 
blades  or  guides  and  the  circumference  of  the 
circle,  these  lengths  may  be  reduced  by  increas- 
ing the  diameters  of  wheels  and  diaphragms 
toward  the  exhaust  end  of  the  turbine,  and 
this  is  sometimes  done. 

11.  Reaction  Turbines. — Although  a  single- 
stage  reaction  turbine  is  possible,  all  practical 
reaction  turbines  are  compound  or  multi-stage. 

There  are  no  nozzles  in  the  ordinary  sense  in  which  the  term  is  used,  and 
steam  enters  around  the  entire  circumference  through  the  usual  form  of 
guides.  Unlike  the  turbines  previously  considered,  there  is  a  drop  of 
pressure  and  increase  of  volume  and  relative  velocity,  throughout  all 
guides  and  blades,  which  are  proportioned  to  provide  for  it.  The  blades 
are  attached  to  a  drum  instead  of  a  disc  such  as  is  generally  used  for 
impulse  turbines,  and  increase  in  length  toward  the  exhaust  end  of  the 


FIG. 


Abs.Pres. 


26.  —  Pressure-stage    im- 
pulse turbine. 


28 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


turbine  to  accommodate  the  increasing  volume.     As  there  are  a  large 
number  of  stages,  the  pressure  drop  is  small  in  each  stage,  which  simplifies 

matters  from  the  standpoint  of  construc- 
tion as  will  be  shown  in  Chap.  XV. 

Figure  27  shows  diagrams  for  four 
stages  of  a  reaction  turbine.  The  ends  of 
blades  and  guides,  more  usually  open  in 
this  type,  are  shown  closed,  or  shrouded, 
according  to  the  practice  of  at  least  one 
builder  in  this  country. 

To  simplify  construction,  blade  lengths 
actually  increase  by  groups,  two  to 
eighteen  rows  of  blades  to  each  group, 
which  is  called  a  barrel.  A  glance  at  the 
steam  tables  shows  that  the  volume  in- 
creases very  rapidly  as  pressure  decreases, 
so  that  where  high  vacuum  is  employed, 
the  blade  lengths  would  be  excessive  if  the 
drum  diameter  were  constant;  to  obviate 
this,  the  diameter  is  increased  toward 


(r^y Pressure 
FIG.  27. — Reaction  turbine. 


the  low-pressure  end,  allowing  shorter  blades  for  the  required  steam 
passages.  Each  portion  of  the  drum  of  a  different  diameter  is  called 
a  cylinder  and  there  are  usually  from  two  to  five  barrels  on  each  cyl- 
inder. This  is  shown  in  Fig.  28.  Due  to  the  higher  peripheral  velocities 


FIG.  28. — Reaction  turbine. 

of  the  larger  cylinders,  the  velocity  of  steam  flow  increases  with  the  size 
of  the  cylinder. 

The  steam  pressure  acting  on  the  annular  spaces  due  to  increasing  the 


THE  STEAM  TURBINE  29 

drum  diameter  causes  an  end-thrust  on  the  turbine  shaft.  This  is  bal- 
anced as  shown  by  discs  at  the  left,  of  the  same  diameter  as  the  differ- 
ent cylinders,  and  connected  respectively  with  them  by  steam  passages. 
12.  Combinations. — The  turbine  types  set  forth  in  the  preceding  para- 
graphs represent  either  separately  or  in  combination,  practically  all 
successful  turbines  built  in  the  United  States.  Some  slight  modifica- 
tions which  may  be  made  as  a  compromise  between  practical  conditions 
will  be  considered  in  Chap.  XV,  but  in  general,  these  types  may  be  said  to 
include  the  elements  of  steam  turbine  design.  These  are  separated 
below,  and  inasmuch  as  the  term  compound  is  now  much  used  in  a  broader 
sense  to  designate  unit  design,  it  will  be  omitted  in  connection  with  the 
elementary  turbines. 

A.  Single-stage  impulse. 


Elements 


B.  Velocity-stage  impulse. 
I  C.  Pressure-stage  impulse. 
[  D.  Reaction  (multi-stage). 


Double-flow  Turbine.  —  The  blading  may  be  so  arranged  that  steam  is 
admitted  near  the  center  of  the  shaft  and  flows  toward  both  ends  where  it 
is  exhausted.  Such  a  turbine  is  known  as  a  double-flow  turbine,  and  may 
be  either  of  the  impulse  or  reaction  type  or  a  combination  of  the  two. 

Pressure-velocity-stage  Turbine.  —  This  consists  of  a  pressure-stage  tur- 
bine with  two  or  more  velocity  stages  in  one  or  more  of  the  pressure 
stages.  The  Curtis  turbine  as  formerly  built  by  the  General  Electric 
Company,  and  still  made  in  the  smaller  units,  is  perhaps  the  best  known 
example.  There  are  from  two  to  five  pressure  stages  with  one  or  two  ve- 
locity stages  per  pressure  stage.  Assuming  a  simple  impulse  turbine  as 
in  Pars.  9  and  10,  running  12,000  r.p.m.,  a  combination  of  the  calculations 
of  those  paragraphs  would  give  for  a  4-stage  turbine  with  two  wheels  per 
stage, 

12,000 


The  Curtis  marine  turbine,  built  by  the  Fore  River  Shipbuilding 
Corporation,  has  a  varying  number  of  velocity  stages;  such  a  turbine  is 
illustrated  and  described  by  Prof.  Peabody  in  his  book,  The  Steam  Turbine, 
in  which  the  first  pressure  stage  has  four  rows  of  moving  blades,  the  second 
to  the  sixth  stages  three  rows,  and  each  of  the  ten  stages  following,  two 
rows.  The  first  stage  of  some  turbines  have  two  velocity  stages  and  the 
rest  one  stage;  these  are  sometimes  known  as  the  Curtis-Rateau  type. 

Repeated-flow  turbines  are  velocity-stage  turbines  using  a  single  row 
of  moving  blades.  The  steam  from  the  nozzles  having  given  up  part  of  its 


30  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

velocity  to  the  blades,  is  redirected  to  the  wheel.  This  method  is  usually 
confined  to  small  turbines. 

Impulse-and-reaction  Turbine. — The  first  stage  of  this  turbine  contains 
an  impulse  wheel,  or  in  some  designs,  there  may  be  several  impulse  pres- 
sure stages  with  a  varying  number  of  velocity  stages  per  stage.  The  ex- 
haust from  these  then  enters  a  reaction  turbine,  generally  on  the  same 
shaft,  where  the  remainder  of  the  work  is  done.  These  turbines  are  some- 
times known  as  the  disc-and-drum  type,  the  impulse  wheels  being  disc 
wheels,  while  the  reaction  blading  is  on  a  drum,  according  to  usual  reac- 
tion-turbine practice.  The  reaction  portion  of  the  turbine  is  sometimes  of 
the  double-flow  design. 

Due  to  full  peripheral  admission  in  the  reaction  turbine,  the  blading 
at  the  high-pressure  end  is  very  short,  and  due  to  the  low  velocity,  and 
expansion  during  the  passage  of  steam  through  the  blading,  the  tip  leakage 
is  comparatively  large.  The  leakage  from  this  cause  is  much  less  in  im- 
pulse blading,  which  may  also  have  partial  admission  and  greater  length, 
therefore  the  impulse  principle  is  best  adapted  for  the  high-pressure  end 
of  the  turbine,  while  for  low  pressures  the  reaction  turbine  is  satisfac- 
tory, and  considered  by  some  to  be  more  efficient. 

Low-pressure  Turbines. — These  may  be  of  any  type,  designed  to 
receive  low-pressure  steam,  such  as  the  exhaust  from  reciprocating  engines. 
Extensions  to  reciprocating  engine  plants  have  been  made  in  this  way,  in- 
creasing both  output  and  economy,  but  new  plants  will  probably  not  be 
designed  in  this  way:  in  fact,  a  large  reciprocating  plant  was  recently  re- 
placed by  turbine  power  when  the  demand  for  increased  power  was  made. 

Mixed-pressure  Turbines. — These  may  be  turbines  of  any  type,  single 
or  combined,  arranged  to  receive  lower-pressure  steam  at  one  of  the  lower 
stages.  They  may  operate  as  low-pressure  turbines,  as  high-pressure  tur- 
bines, or  with  a  combination  of  high  and  low  pressures. 

Bleeder  turbines  are  so  designed  that  steam  may  be  withdrawn  for 
heating  or  some  industrial  process  from  one  of  the  lower  stages.  The 
stages  below  that  from  which  the  steam  is  taken  are  designed  to  operate 
with  a  decreased  weight  of  steam. 

Compound  Turbine. — While  all  but  the  single-stage  impulse  turbines 
are  compound  in  a  sense,  the  present  classification  refers  to  the  division 
of  the  turbine  into  two  or  more  elements.  The  compound  reaction 
marine  turbine  is  sometimes  composed  of  three  elements,  one  high-pressure 
and  two  low-pressure.  These  are  on  separate  shafts,  forming  a  cross-com- 
pound turbine.  Several  large  units  of  large  power  have  recently  been 
built  of  the  cross-compound  type  with  one  high-pressure  and  one  low-pres- 
sure element.  A  large  tandem-compound  turbine  with  high-pressure  and 


THE  STEAM  TURBINE  31 

low-pressure  elements  in  separate  casings  but  on  the  same  shaft,  was  re- 
cently installed.  It  is  of  the  Rateau  or  pressure-stage  type.  The  high- 
pressure  element  has  ten  pressure  stages,  there  being  two  velocity  stages 
in  the  first  stage.  The  low-pressure  element  is  a  double-flow  Rateau 
turbine  with  two  pressure  stages  on  either  side. 

Condensing  and  noncondensing  turbines  are  both  used.  As  further 
explained  in  Chap.  XV,  the  influence  of  the  condenser  on  capacity  and 
economy  is  much  greater  in  the  turbine  than  in  the  reciprocating  engine, 
so  that,  except  for  some  special  cases  and  for  small  powers,  turbines 
are  usually  condensing. 

Some  of  the  advantages  of  the  different  combinations  will  be  men- 
tioned in  Chap.  XV.  Other  arrangements  possessing  merit  may  probably 
be  made,  and  as  with  the  steam  engine,  the  adoption  of  any  one  design 
seems  rather  remote.  The  use  of  the  different  elemental  types  permits 
considerable  flexibility  of  design,  which  is  probably  desirable. 

Governing. — Steam  turbines  are  governed  by  throttling  or  by  admit- 
ting steam  to  a  varying  number  of  nozzles  according  to  the  load.  The 
control  of  steam  to  the  first  stage  is  sufficient  for  the  usual  range  of  load : 
when  large  overloads  are  imposed,  the  governor  admits  high-pressure 
steam  to  some  lower  stage.  In  addition  to  the  regular  governor,  practi- 
cally all  turbines  are  furnished  with  an  emergency  governor  which,  when 
a  certain  speed  above  normal  is  reached,  automatically  closes  the  throttle. 

The  speed  of  steam  turbines,  except  when  reducing  gears  are  used,  is 
dependent  upon  the  machinery  driven,  which  usually  necessitates  a  multi- 
stage turbine  of  either  the  pressure-stage-impulse,  or  the  reaction  type. 

References 

Westinghouse  45,000  kw.  Cross-Compound  Turbine.     Power,  April  18,  1916. 
Exhaust  Steam  Turbines.    Power,  June  6,  1916. 


CHAPTER  V 
THE  INTERNAL-COMBUSTION  ENGINE 

13.  On  account  of  its  superior  relative  thermal  efficiency,  especially 
in  the  smaller  units,  the  internal-combustion  engine  has  been  attractive 
since  its  first  really  practical  introduction  by  Dr.  Otto  in  1876.     The 
obstacles  to  reliable  operation  have  gradually  been  overcome  by  persist- 
ent designers  and  experimenters  until  today,  there  seems  to  be  but  few 
power  problems  where  it  may  not  be  used  as  an  alternative  to  the  steam 
engine.     For  automobile,  launch,  small  yacht  and  aerial  propulsion  it 
seems  to'  hold  an  almost  undisputed  field,  and  no  other  form  of  motor  is 
suitable  for  submarine  service.     Internal-combustion  reversing  engines  of 
considerable  power  have  been  successfully  applied  to  marine  propulsion. 
Large  gas  engines  have  been  popular  in  Europe  and  have  been  used  to 
some  extent  in  this  country,  notably  in  connection  with  blast  furnaces 
where  the  waste  gas  is  used  as  fuel,  thus  effecting  great  economy.     In 
small  and  medium  powers,  the  internal-combustion  engine  is  adapted  to 
nearly  all  classes  of  service. 

Auxiliary  apparatus  is  not  essential  to  the  thermal  cycle  of  an  inter- 
nal-combustion engine.  With  steam,  isothermal  expansion  occurs  in  the 
boiler  and  piping  as  well  as  in  the  engine  cylinder,  and  for  the  condens- 
ing steam  engine,  isothermal  compression  occurs  partly  in  the  piping  and 
condenser;  but  with  the  internal-combustion  engine,  the  cycle  is  complete 
within  the  engine,  and  aside  from  the  manufacture  or  preparation  of  the 
fuel,  which  is  not  a  part  of  the  cycle,  no  heat  transfer,  expansion  or  com- 
pression occurs  outside  the  engine  cylinder.  The  mechanism  concerned 
with  the  supply  and  ignition  of  fuel  will  be  treated  under  valve  gears  in 
Chap.  XX,  but  no  auxiliaries  analogous  to  boiler,  condenser,  etc.  will  be 
considered. 

14.  Classification  and  Cycles. — Many  of  the  classifications  given  in 
Chap.   Ill   for  the   steam   engine    apply   to   the   internal-combustion 
engine,  and  as  it  is  obvious  when  they  do  not,  no  repetition  is  deemed 
necessary. 

All  practical  internal-combustion  engines  operate  with  either  four  or 
two-strokes  per  cycle  and  are  known  respectively  by  the  abbreviated  terms 
four-cycle  engines  and  two-cycle  engines.  Practically  all  types  are,  or 

32 


THE  INTERNAL-COMBUSTION  ENGINE  33 

may  be,  built  upon  either  of  these  cycles.  The  thermal  cycle  is  really 
the  same  in  each  case,  the  difference  being  in  the  mode  of  receiving  the 
fresh  charge  and  exhausting  the  products  of  combustion. 

Internal-combustion  engines  are  also  classified  by  the  manner  of  burn- 
ing the  fuel.  If  the  fuel  is  burned  suddenly  with  an  explosion,  the  piston 
movement  is  so  slight  during  combustion  that  the  volume  is  practically 
constant;  the  cycle  is  therefore  known  as  the  constant-volume  cycle.  If 
combustion  is  slower,  so  that  the  pressure  remains  nearly  constant  as  the 
piston  moves  away  from  the  end  of  the  stroke  until  combustion  is  com- 
plete, the  cycle  is  the  constant-pressure  cycle. 

If  an  engine  uses  gaseous  fuel  it  is  known  as  a  gas  engine — although  this 
term  is  often  applied  to  all  internal-combustion  engines — if  liquid  fuel 
is  used,  it  is  known  as  an  oil  engine.  Engines  are  also  commonly  known 
by  the  names  of  the  particular  gas  or  oil  they  use,  as  producer-gas  engine, 
gasoline-  engine,  kerosene  engine  or  alcohol  engine. 

The  following  classification  will  cover  all  types  treated  in  this  book: 


Gas 
( Constant-volume 


4-cycle  or  2-cycle 


j  Light-oil. 
1  Heavy-oil. 


Constant-pressure—Diesel. 


The  4-stroke,  constant-volume  cycle  was  the  one  employed  by  Dr.  Otto 
and  is  commonly  known  as  the  Otto  cycle,  while  the  4-stroke,  constant- 
pressure  cycle  was  the  practical  cycle  of  Dr.  Diesel  and  is  known  as  the 
Diesel  cycle;  but  it  is  not  uncommon  for  the  2-stroke  cycle  for  these  two 
methods  of  combustion  to  be  also  thus  designated.  The  Otto  and  Diesel 
cycles  are  fully  described  in  Chap.  VI,  Pars.  25  to  30,  but  a  general 
description  of  the  4-stroke  and  2-stroke  cycles  will  now  be  given. 

Four-cycle  Engine. — In  Fig.  29  are  diagrams  showing  the  position  of 
the  inlet  and  exhaust  valves  for  the  four  strokes  of  the  cycle.  Above 
are  shown  conventional  indicator  diagrams  for  the  Otto  and  Diesel 
cycles.  There  are  really  six  steps  in  the  cycle  as  follows: 

1.  Suction. — Starting  from  the  position  at  the  head  end  of  the  cylinder 
shown  by  the  dotted  lines,  with  the  inlet  valve  open  and  exhaust  valve 
closed  as  shown  in  Fig.  29-A,  the  piston  moves  to  the  right.  A  mixture 
of  fuel  and  air,  or  with  some  engines,  air  alone,  enters  the  inlet  valve, 
which  is  either  held  open  against  the  pressure  of  a  light  spring  by  suction, 
or  against  a  heavier  spring  by  a  cam-operated  mechanism.  This  stroke 
is  indicated  by  the  line  1-2  on  the  indicator  diagrams.  At  the  end  of  the 
stroke  the  inlet  valve  closes. 


34 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


2.  Compression. — With  both  valves  closed,  the  piston  now  moves  to 
the  left  as  shown  in  Fig.  29-B,  compressing  the  charge.     This  is  shown  on 
the  indicator  diagrams  by  line  2-3. 

3.  Ignition  now  takes  place  along  line  3-4  of  the  indicator  diagrams. 
This  is  assumed  instantaneous  for  the  constant-volume  engine  and  is 
caused  by  an  electric  spark,  or  by  the  heat  of  compression  to  be  explained 


FIG.  29. — Four-stroke  cycle. 

presently.  If  air  alone  enters  by  the  inlet  valve  during  the  suction  stroke, 
as  in  the  heavy-oil  engines,  oil  is  admitted  to  the  cylinder  during  the 
same  stroke,  or  is  now  sprayed  into  the  end  of  the  cylinder,  against  a 
hot  plate  or  into  a  hot  bulb  if  the  constant  -volume  cycle  is  used.  With 
either  hot  bulb  or  hot  plate,  combustion  is  assumed  instantaneous.  With 
the  Diesel  cycle,  oil  is  sprayed  into  the  heated  air  and  burns  more  slowly 


THE  INTERNAL-COMBUSTION  ENGINE 


35 


at  nearly  constant  pressure,  while  a  small  portion  of  the  stroke  is  ac- 
complished by  the  piston. 

4.  Expansion. — With  both  valves  closed  as  in  Fig.  29-C,  the  piston 
moves  to  the  right  as  the  pressure  falls,  drawing  line  4-5  of  the  indicator 
diagrams. 

5.  Exhaust. — The    exhaust   valve,    always    mechanically    operated, 
opens,  and  the  pressure  drops  along  line  5-2. 

6.  Exhaust  Stroke. — With  the  exhaust  valve  open  as  in  Fig.29  -D,  the 
piston  moves  to  the  left,  drawing  line  2-1  and  expelling  the  products  of 
combustion.     This  completes  the  cycle. 


CONSTANT  PRESSURE 


FIG.  30. — Two-stroke  cycle. 

In  practice,  the  valve  openings  and  ignition  do  not  occur  exactly  at 
the  ends  of  the  stroke;  neither  is  combustion  instantaneous  in  constant- 
volume  engines.  This  is  treated  in  Chap.  XX.  Actual  indicator 
diagrams  also  differ  somewhat  from  the  conventional.  An  actual  con- 
stant-volume diagram  is  shown  in  Fig.  34  and  a  constant-pressure  diagram 
is  shown  in  Fig.  38. 

Two-cycle  Engine. — In  this  engine  the  suction  and  exhaust  strokes  are 
eliminated.  There  are  no  inlet  and  exhaust  valves  in  the  cylinder,  and 
charging  and  exhaust  are  accomplished  by  means  of  ports  in  the  cylinder 


36  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

wall  uncovered  at  the  proper  time  by  the  piston,  which  performs  the  func- 
tion of  a  valve.  Figure  30  is  a  diagram  of  a  2-cycle  engine,  with  in- 
dicator diagrams  for  the  constant-volume  and  constant-pressure  cycles. 

When  combustion  is  complete  at  1  on  the  indicator  diagrams,  the 
piston  moves  to  the  right  and  the  pressure  drops  during  expansion, 
drawing  the  curve  1-2.  At  the  point  2,  shown  by  the  full-line  piston 
position,  the  exhaust  port  (a)  begins  to  open.  As  the  piston  moved  to 
the  right  it  compressed  the  fresh  charge  of  air  and  fuel  (or  air  alone  for 
Diesel  and  other  heavy-oil  engines)  in  the  crank  case  (6) .  As  the  exhaust 
port  (a)  is  opened,  the  burnt  gases  rush  out  and  the  pressure  in  the 
cylinder  drops.  A  little  further  movement  of  the  piston  to  the  right 
opens  the  inlet  port  (c)  connected  with  the  crank  case,  and  the  fresh 
charge  under  light  pressure  rushes  into  the  cylinder,  is  deflected  by  the 
projection  (d)  on  the  piston  so  that  it  may  not  pass  across  the  cylinder 
and  out  of  the  exhaust  port  (a).  This  deflected  stream  aids  in  the 
expulsion  of  the  burnt  charge. 

During  the  recharging  period  the  piston  moves  to  its  extreme  right 
position,  completely  uncovering  both  ports  as  shown  by  the  dotted  lines, 
and  completing  line  2-3,  which  is  arbitrarily  shown  as  a  straight  line 
for  convenience  of  illustration.  The  pressure  in  crank  case  and  cylinder 
at  this  point  is  presumably  about  the  same,  and  equal  to  that  of  the 
atmosphere,  and  there  is  no  noticeable  change  until  the  piston  on  its 
return  stroke  closes  the  exhaust  port  (a),  when  compression  begins  and 
the  line  4-5  is  drawn  on  the  indicator  diagrams.  During  this  stroke  the 
fresh  charge  is  drawn  into  the  crank  case  through  the  check  valve  (e), 
due  to  the  partial  vacuum  formed  by  the  movement  of  the  piston. 

Ignition  now  occurs,  and  with  certain  heavy-oil  engines  the  fuel  is 
sprayed  into  the  cylinder  exactly  as  with  a'4-cycle  engine,  except  that 
it  occurs  every  revolution  instead  of  every  two  revolutions. 

To  follow  the  crank-case  cycle  more  in  detail,  assume  the  piston  to  be 
in  the  extreme  position  to  the  right  at  the  point  8  of  the  crank  case  in- 
dicator diagram.  As  the  piston  moves  to  the  left  it  increases  in  velocity, 
then  decreases  as  it  nears  the  end  of  the  stroke;  the  vacuum  varies  with 
the  velocity  as  shown  by  the  line  8-6  drawn  during  the  stroke  as  the 
charge  is  drawn  in  through  check  valve  (e).  The  crank  case  is  now  full  of 
the  charge  at  approximately  atmospheric  pressure,  which  is  prevented 
from  leaking  away  by  the  check  valve  (e),  and  air-tight  joints  at  all  parts 
of  the  casing.  The  piston  now  moves  to  the  right,  compressing  the 
charge.  As  the  volume  of  the  crank  case  is  much  greater  than  the  volume 
displaced  by  the  piston,  the  compression  pressure  is  not  very  high,  being 
from  5  to  8  Ib.  per  sq.  in.  When  point  7  of  the  compression  line  is  reached, 


THE  INTERNAL-COMBUSTION  ENGINE 


37 


the  piston  opens  the  inlet  port;  the  charge  rushes  in  as  previously  de- 
scribed and  the  pressure  drops  from  7  to  8  as  the  piston  reaches  the  end  of 
the  stroke,  completing  the  crank-case  cycle. 

The  work  done  in  the  crank  case  is  negative  work  and  must  be  de- 
ducted from  the  work  done  in  the  cylinder  as  shown  by  an  indicator 
diagram  taken  from  it. 

A  study  of  the  valve  gear  of  the  4-cycle  engine  given  in  Chap.  XX 
shows  that  it  will  run  in  but  one  direction  unless  some  special  reversing 
mechanism  is  employed,  while  it  is  obvious  from  Fig.  31  that  the  2-cycle 
engine  will  run  in  whichever  direction  it  is  started. 

Fig.  31  shows  a  modification  of  the  2-cycle  engine  known  as  the  3-port 
engine,  in  which  the  check  valve  of  Fig.  30,  admitting  the  charge  to  the 
crank  case,  is  replaced  by  the  port  (g),  which  is  opened  by  the  piston 
as  it  nears  the  end  of  the  stroke.  The  suction  is  increased  up  to  the  time 


F}G.  31. — Three-port  engine. 

of  the  port  opening,  probably  increasing  the  negative  work,  and  the 
charging  of  the  crank  case  must  occur  during  the  period  of  port  opening, 
which  is  less  than  for  the  2-port  engine. 

The  diagrams  given  are  for  single-acting  engines  in  which  the  piston 
performs  the  function  of  a  crosshead.  A  crosshead  must  be  used  with  a 
double-acting  engine,  and  is  sometimes  used  with  single-acting  engines. 
When  used  with  single-acting  2-cycle  engines,  the  crank  end  of  the  cylin- 
der is  enclosed  and  utilized  instead  of  the  crank  case  for  compression  of 
the  charge,  leakage  around  the  piston  rod  being  prevented  by  a  stuffing 
box.  Double-acting  2-cycle  engines  of  large  power,  using  gaseous  fuel, 
are  provided  with  separate  charging  pumps  for  gas  and  air,  operated 
from  the  engine  crank  shaft. 

Cylinder  Cooling. — This  is  a  necessity  with  the  internal-combustion 
engine,  due  to  the  extremely  high  temperatures  developed  in  the  cylinder. 
Without  cooling,  lubrication  would  be  impossible  and  the  metal  would  be 


38  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

weakened.  For  this  purpose,  water,  or  in  some  small  engines  air,  is  cir- 
culated around  the  cylinder  wall,  withdrawing  a  portion  of  the  heat  gener- 
ated by  combustion.  This  will  be  further  mentioned  in  Par.  21. 

Compression  pressure  varies  with  the  fuel  and  method  of  opera- 
tion, and  will  be  considered  in  Chap.  XIV. 

With  this  general  description  of  the  essentials  of  internal-combustion 
engines,  some  differentiating  features  will  be  mentioned  concerning  the 
ultimate  subdivisions  given  early  in  this  paragraph,  viz.,  gas,  light-oil, 
heavy-oil  and  Diesel  engines. 

15.  Gas  engines  use  fuels  which  are  in  the  form  of  a  fixed  gas.     They 
may  be  either  4-cycle,  2-cycle,  single-acting  or  double-acting.     They  are 
always  constant-volume  engines,   no  successful  constant-pressure  gas 
engine  having  yet  been  developed.     They  are  built  for  a  wide  range  of 
power.     The  fuels  used  are  natural  gas,  the  different  forms  of  manufac- 
tured illuminating  gas,  producer  gas  and  blast-furnace  gas.     Ignition  is 
by  electric  spark. 

Natural  gas  is  a  good  fuel  but  is  available  in  but  few  localities. 

Illuminating  gas  is  too  expensive  for  general  use  and  is  only  used  for 
small  powers. 

Producer  gas  is  formed  by  the  partial  combustion  of  fuel.  A  large 
variety  of  fuels  may  be  used,  including  many  waste  products,  but  anthra- 
cite coal  is  perhaps  the  most  used  and  most  satisfactory  for  power  pur- 
poses. Producer  gas  is  much  cheaper  than  illuminating  gas  due  to  less 
expensive  methods  of  manufacture,  and  is  a  very  satisfactory  fuel. 

Blast-furnace  gas,  as  the  name  implies,  is  a  product  of  the  blast  fur- 
nace. It  is  burned  with  difficulty  under  steam  boilers,  but  under  high 
compression  pressure  makes  a  satisfactory  gas-engine  fuel. 

16.  Light-oil  engines  are  either  4-cycle  or  2-cycle,  but   are   practi- 
cally always  single-acting  engines.     They  are  constant-volume  engines, 
although  light  oils  may  be  used  with  constant-pressure  engines.     They  are 
comparatively  small  but  have  a  wide  range  of  application.     The  fuels 
commonly  used  are  gasoline,  kerosene,  distillates — petroleum  products 
between  gasoline  and  kerosene — and  alcohol.     An  engine  built  for  gaso- 
line may  burn  any  of  these  other  fuels,  but  better  results  are  obtained 
by  some  modification;  that  is,  the  compression  pressure  should  be  lower 
for  kerosene  and  higher  for  alcohol,  and  special  devices  for  handling  the 
fuel  are  sometimes  employed.     Air  is  mixed  in  a  carbureter,  the  fuel 
being  introduced  in  the  form  of  a  fine  spray.     Both  air  and  fuel  are  drawn 
through  the  carbureter  by  the  partial  vacuum  formed  during  the  suction 
stroke  of  the  4-cycle  engine,  or  the  pump  stroke  of  the  2-cycle  engine. 
Ignition  is  by  electric  spark. 


THE  INTERNAL-COMBUSTION  ENGINE  39 

17.  Heavy-oil  engines  of  the  constant- volume  type  may  use  any 
fuel  from  kerosene  to  the  heavy  low-grade  oils;  usually  the  heavier  oils 
are  used,  and  seem  to  be  more  satisfactory  even  if  the  question  of  economy 
is  not  considered.  These  engines  are  practically  always  single-acting. 
Ignition  is  due  to  the  heat  of  compression,  no  electric  spark  being  required. 

The  hot-bulb  engine  is  one  of  the  earlier  forms  of  the  heavy-oil  engine. 
The  end  of  the  cylinder  of  one  of  the  older  designs  is  shown  in  Fig.  32. 
The  bulb-shaped  portion  of  the  clearance  space  is  separated  from  the 
cylinder  end  by  a  small  passage.  In  this  bulb  are  a  series  of  projecting 
plates  or  points.  To  start  the  engine  the  bulb  is  brought  to  a  dull-red 
heat  by  means  of  a  torch  and  blower,  but  is  thereafter  kept  hot  by  the 
combustion  of  the  fuel.  This  engine  is  a  4-cycle 
engine,  and  during  the  suction  stroke,  while  pure 
air  is  being  drawn  into  the  cylinder,  oil  is 
sprayed  into  the  bulb  where  it  is  vaporized  by 
the  heat  stored  in  the  metal  walls  and  projecting 
plates.  As  only  burnt  gas  was  left  in  the  bulb  at  the  FlG  32  _Hot  bulb 
end  of  the  previous  exhaust  stroke,  the  oil  vapor  will 
not  ignite.  Should  there  be  air  introduced  at  this  time,  the  temperature 
is  not  equal  to  that  required  for  ignition.  When  air  is  compressed  the 
temperature  rises,  so,  as  the  piston  starts  upon  the  compression  stroke  the 
temperature  gradually  rises;  at  the  same  time,  the  air  is  being  forced  back 
into  the  hot  bulb  and  mixes  with  the  vaporized  oil.  The  projecting 
plates  become  incandescent,  and  at  the  proper  point  in  the  stroke,  deter- 
mined by  experiment,  the  mixture  and  temperature  are  such  that  ignition 
occurs.  The  compression  pressure  is  comparatively  low  in  this. type. 

The  newer  type  of  hot-bulb  engine  and  the  hot-plate  engine  are  some- 
times called  medium-compression  oil  engines.  In  these,  compression  is 
higher,  sometimes  300  Ib.  per  sq.  in.,  and  the  oil  is  sprayed  into  the 
bulb  or  against  the  plate  at  the  end  of  the  compression  stroke  as  in  the 
Diesel  engine.  The  temperature  attained  is  not  high  enough  to  insure 
combustion  without  the  aid  of  the  bulb  or  plate,  which  is  made  red-hot 
by  the  heat  of  compression  added  to  the  heat  of  combustion  of  the  pre- 
vious cycle.  Combustion  is  theoretically  at  constant-volume,  the  pres- 
sure rising  to  between  450  and  500  Ib.  per  sq.  in.,  gage.  It  is  probable 
that  combustion  continues  along  the  stroke  for  a  short  distance.  While 
usually  a  4-cycle  engine,  the  2-cycle  principle  may  be  applied  to  this  type. 

These  engines  are  often  known  as  semi-Diesel  engines,  although  it  is 
claimed  by  some  that  this  name  applies  only  to  engines  in  which  the  fuel 
is  directly  injected  without  the  use  of  air;  there  seems  to  be  little  reason 
for  this  distinction. 


40  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

18.  The  Diesel  engine  is  the  practical  representative  of  the  constant- 
pressure  type  and  has  attained  the  highest  thermal  efficiency  of  any  heat 
engine.    For  the  same  compression  pressure,  it  will  be  seen  from  Chap.  VI, 
Par.  26,  that  the  theoretical  constant-volume  cycle  has  a  higher  efficiency 
than  the  constant-pressure  cycle,  but  the  absence  of  fuel  during  com- 
pression permits  a  much  higher  compression  in  the  Diesel  cycle,  offsetting 
the  theoretical  advantage  of  the  Otto  cycle.     The  temperature  due  to  this 
high  pressure — 450  to  600  Ib.  gage — is  high  enough  to  ignite  the  fuel  as 
fast  as  it  enters  the  cylinder  without  the  aid  of  hot  bulb  or  plate.     The 
fuel  is  forced  in  by  air  pressure  much  higher  than  the  compression  pres- 
sure, the  design  of  spray  nozzle  and  timing  of  discharge  being  such  that 
combustion  is  at  approximately  constant  pressure.     In  most  cases,  air 
for  fuel  injection  is  provided  by  a  compressor  operated  from  the  engine 
shaft,  and  stored  in  high-pressure  tanks  or  bottles ;  but  sometimes  the  air 
is  led  directly  from  the  compressor  to  the  spray  nozzle,  a  definite  amount 
being  injected  with  each  charge.     In  some  engines  the  fuel  is  injected 
directly  from  an  oil  pump. 

While  the  Diesel  engine  proper  is  a  4-cycle  engine,  the  2-cycle  prin- 
ciple is  successfully  applied. 

These  engines  have  been  thus  far  single-acting  engines,  and  burn  the 
same  fuels  as  the  heavy-oil  constant-volume  engines.  They  may  even 
burn  oils  harder  to  ignite,  and  according  to  E.  W.  Roberts  in  "The Gas 
Engine"  of  July,  1916  (Liquid  Fuels,  Present  and  Future),  they  may  burn 
cottonseed  oil,  olive  oil,  cocoanut  oil  and  mustard-seed  oil,  the  last  named 
having  been  used  in  India  twenty  years  ago. 

Diesel  engines  are  usually  built  for  medium  powers,  although  engines 
developing  over  3000  b.h.p.  are  now  in  use.  While  usually  a  heavy  engine 
due  to  the  high  pressure,  the  capacity  is  greater  for  a  given  cylinder 
diameter,  and  it  is  not  unlikely  that  the  2-cycle  Diesel  engine  will  be 
developed  for  automobile  propulsion  and  even  for  the  airplane. 

19.  Governing. — The  regulation  of  small  stationary  engines  using 
gas  or  light  oils  is  often  accomplished  by  the  "hit-and-miss"  governor; 
that  is,  when  less  power  is  required,  one  or  more  explosions  are  missed. 
This  may  be  accomplished  in  different  ways,  but  when  the  automatic 
inlet  valve  is  used,  the  exhaust  valve  is  held  open  during  the  cycles  when 
the  explosion  is  to  be  missed.     This  relieves  the  suction,  and  the  inlet 
valve  is  not  drawn  open.     This  method  is  simple,  and  as  adjustment 
may  be  made  to  give  the  best  fuel  mixture  and  compression  at  all  times, 
the  engine  always  operates  with  the  best  thermal  efficiency  if  the  load 
is  not  so  light  as  to  unduly  cool  the  cylinder. 

For  large  engines  using  these  fuels,  governing  by  throttling  is  now 


•  THE  INTERNAL-COMBUSTION  ENGINE  41 

the  most  common  method.  This  is  sometimes  applied  to  the  fuel  only, 
especially  with  gas  engines.  This  changes  the  proportion  of  air  and  gas 
and  is  known  as  quality  governing.  More  usually  the  mixture  is  throttled 
as  it  passes  into  the  cylinder,  the  ratio  of  gas  to  air  being  kept  as  nearly 
constant  as  possible:  this  is  the  quantity  method.  The  governing 
mechanism  is  more  simple  for  quantity  governing,  and  while  the  theoret- 
ical thermal  cycle  efficiency  is  greater  for  the  quality  method  due  to 
the  constant  best  compression  pressure,  the  better  combustion  of  a 
constant  mixture  more  than  offsets  this,  according  to  several  leading 
authorities. 

A  combination  of  the  quality  and  quantity  methods  is  sometimes 
used.  As  lean  mixtures  (with  a  small  percentage  of  gas)  do  not  ignite 
easily,  and  burn  slowly,  a. minimum  satisfactory  ratio  of  gas  to  air  is 
handled  by  the  quality  method;  when  more  power  is  required,  more 
gas  is  admitted  and  when  less  power  than  that  given  by  this  minimum 
mixture,  the  entire  charge  is  throttled.  Although  the  governing  mech- 
anism is  somewhat  more  complicated  this  method  gives  good  results. 

Variable-speed  engines  such  as  those  used  for  automobiles,  govern  by 
changing  the  mixture,  by  throttling  and  by  timing  the  ignition,  all  by 
hand  control. 

In  the  heavy-oil  engine,  governing  is  effected  by  changing  the  amount 
of  oil  pumped  to  the  spray  nozzle.  The  pump  handles  more  oil  than  is 
required  for  the  maximum  load,  and  that  not  required  is  by-passed  by 
means  of  the  governor  mechanism. 

Governors  and  their  connections  will  be  treated  in  Chaps.  XIX  and 
XX. 

20.  Starting. — Small  engines  are  commonly  started  by  hand  after 
properly   adjusting   the  fuel   valve   and  ignition   apparatus.     Electric 
starters  are  used  for  automobile  engines,  the  current  for  which  is  furnished 
by  a  storage  battery  which  is  charged  while  the  engine  is  running. 

Large  engines  are  usually  started  by  compressed  air  admitted  to  the 
cylinder  by  separate  cams.  When  the  engine  is  up  to  speed,  fuel  is 
turned  on  and  the  regular  cycle  begins.  With  the  Diesel  and  semi- 
Diesel  engines,  air  for  starting  is  commonly  furnished  by  the  compressor 
which  furnishes  the  air  for  fuel  injection.  It  is  taken  from  the  lower 
stage  of  the  compressor  and  stored  in  a  separate  tank,  the  pressure  in 
which  is  automatically  governed. 

21.  Cylinder  cooling  apparatus  and  exhaust  mufflers  may  be  con- 
sidered as  auxiliaries,  but  are  so  essential  to  the  operation  of  internal- 
combustion  engines  that  they  will  be  briefly  mentioned.     With  the  ex- 
ception of  a  few  small  engines,  the  cylinder  is  cooled  by  the  circulation 


42 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


of  water.  Water  cooling  may  be  accomplished  by  gravity  or  by  forced 
circulation.  A  gravity  system  is  shown  in  diagram  in  Fig.  33.  As  the 
water  is  heated  in  the  cylinder  it  rises,  flowing  to  the  upper  part  of 
the  tank  where  it  is  cooled  by  exposure  to  the  air.  The  water  falls  to  the 
bottom  of  the  tank  as  it  cools  and  enters  the  engine  cylinder  by  the  pipe 
(6).  As  the  force  causing  the  flow  in  a  gravity  system  is  very  small  and 
the  slightest  stoppage  of  the  pipe  may  cause  it  to  cease  circulating,  a 
pump  is  usually  placed  in  the  system,  making  the  flow  more  positive. 
Where  water  is  scarce,  the  power  great,  or  large  water  storage  not  prac- 
ticable, various  devices  are  used  for  cooling  the  water.  In  the  automobile, 
a  radiator  placed  at  the  front  of  the  car,  is  used,  through  which  air  is 
forced  by  a  fan.  For  large  powers  the  cooling  tower  or  cooling  pond  is 
used,  as  for  the  cooling  of  circulating  water  for  condensers. 


FIG.  33. — Gravity  cooling  system. 

The  exhaust  of  an  internal-combustion  engine  is  noisy  and  disagree- 
able and  is  rightly  considered  a  nuisance.  To  reduce  this  noise  the  muf- 
fler is  used.  The  muffler  is  an  enlargement  of  the  exhaust  pipe  containing 
perforated  plates,  the  object  being  to  reduce  the  velocity  of  the  gas. 
Cooling  is  sometimes  resorted  to,  decreasing  the  volume  of  gas  passed. 
There  are  a  great  variety  of  designs,  but  the  principle  is  the  same  in 
all. 

The  noise  due  to  intake  of  the  charge  is  often  objectionable  in  engines 
of  large  capacity  and  this  is  also  muffled. 

The  remaining  paragraphs  of  this  chapter  will  be  devoted  to  illustra- 
tions and  descriptions  of  a  number  of  engines  built  by  leading  manufac- 
turers. Details  of  these  and  of  certain  auxiliaries  will  be  treated  in 
later  chapters. 


THE  INTERNAL-COMBUSTION  ENGINE 


43 


ILLUSTRATIONS  AND  DATA  FROM  PRACTICE 

22.  The  Bruce-Macbeth  Co.  of  Cleveland,  Ohio,  manufacture  gas 
engines  for  natural  and  producer  gas.  They  are  of  the  4-cycle,  vertical, 
multi-cylinder  type  and  are  suitable  for  a  wide  range  of  service.  They  are 
provided  with  dual  jump-spark  ignition,  receiving  current  from  a  magneto 
located  on  the  cam  shaft. 

Fig.  34  is  an  indicator  diagram  from  a  4-cylinder,  12J^  by  14  in. 
Bruce-Macbeth  engine,  and  Fig.  35  is  a 
gas-consumption  curve  for  the  same   en- 
gine using  natural  gas,  the  rated  capacity 
being  150  b.h.p. 

Table  1  gives  general  information  con- ' 
cerning  Bruce-Macbeth  engines,  including 
net  price,    which   quantity  may  be  sub- 
ject to  fluctuation. 

The  De  La  Vergne  Type  "  FH  "  Crude-oil  Engine  is  built  by  the  De  La 
Vergne  Machine  Co.,  New  York.  These  engines  are  of  the  horizontal,  4- 
cycle,  hot-bulb,  semi-Diesel  type,  liquid  fuel  being  injected  at  the  end  of 
the  compression  stroke  by  means  of  compressed  air.  Ignition  is  due  to 
heat  deposited  in  the  walls  of  the  vaporizer  chamber  or  hot  bulb  by  the 


FIG.    34. — Indicator    diagram 
Bruce-Macbeth  engine. 


from 


C(J 

o: 

=  16 
CD 

•L'2 

§8 

0 

*4 

3 

0 

j 

\ 

\ 

**^X 

^u**^. 

•*-  —  . 

*»•   .• 

»- 

•MH^M 

i^   "~*J"" 

3              40              80               120             160           200 

B.H.P. 

FIG.  35. — Gas  consumption  of  a  Bruce-Macbeth  engine. 


previous  explosion.  The  compression  pressure  is  about  280  Ib.  per  sq.  in. 
gage,  and  causes  a  temperature  which,  while  not  sufficient  to  ignite  the 
charge,  aids  combustion.  The  explosion  pressure  is  about  480  Ib.  gage. 
These  engines  are  of  extremely  rugged  construction  and  of  excellent  design 
throughout.  One  of  them  gave  a  continuous  service  of  800  hours,  as 
stated  in  the  Trans.  A.  S.M.E.  referred  to  at  the  end  of  this  chapter. 


44 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


eaqoui 


saqoui  -BIQ 


saqout 


saqouT  "Bi 


aa^uao  "BT 


"BtQ 


saqout  ui 

JO  "Bi 


saqotn  UT  ajjoJ^g 


saqoui  UT  aaog 


•d-q-q 


•d-q-q 


SP  7  ? 

t-    00 


o   (M 


H 

j  j  j  y 

os  o  ^b  oo 


00s  iff  od"  c< 


(N   (N    10   00 


CO 


o  io 

CO    CO 


iO   CO    00 


(N   CO   CO   Tin 


CO   1>    00 


CO  l>   00 


TH    T^H    CO    00 


O   <M   rfl   00 


CO   Tin"  00^ 


10*0 
l>   (N 


O   iO   iO 

r-l    r-l    <M 


"^    "^    !>•    CO    CO 

j  j  j  j  ^i 

t>   t>-    t>   00   00 


t  T  7  ? 


CO 


CO   CO   CO   rt< 


CO   CO   CO   T}H 


CO   CO    CO    CO   CO 


\N  \eq  \eq 

r-K  i-K   r-i\   COX  W\ 


\oo  \oo 

r-       COX  W\ 
TjH     TjH     1C     IO 


<N   <N    <M   CO   CO 


CO    CO    CO   "# 


CO 


Tfl    Tfl    -^    rH     rH 

CO   CO   CO   "*   Tt< 


Tt<     TjH     TjH     10 


(N   (N   <N  O  O 

iO  iO  >O  CO  CO 


rH    rH     rH    CO    CO 


r-K 
^     O     rH     (N 

O5    O5    rH    rH    rH 


SO    O    IO    IO 
O   O   l>  t^ 
CO    CO    '-'O    C^l    C1! 


THE  INTERNAL-COMBUSTION  ENGINE 


45 


Indicator  diagrams  from  a  Type  "FH"  engine  is  shown  in  Fig.  36. 

The  fuel  consumption  of  a  20  by  34^  in.  DeLa  Vergne  Type  "FH" 

engine  is  given  in  Fig.  37.     This  engine  is  rated  at  140  b.h.p.  at  165r.p.m. 

The  Mclntosh  and  Seymour  Diesel  Type  Oil  Engines  are  built  by  the 


Full  Load 

FIG.  36. — Indicator  diagrams  from  a  De  La  Vergne  type  "FH"  engine. 

Mclntosh  and  Seymour  Corporation,  Auburn,  N.  Y.  They  are  typical 
vertical,  4-cycle  Diesel  engines  having  the  fuel  injected  under  air  pressure. 
They  are  built  in  two  styles,  Type  A  having  four  cylinders  and  individual 
frames  for  each  cylinder,  bolted  to  a  common  base;  and  Type  B  having 
from  one  to  six  cylinders  bolted  to  a  single  box  frame.  Both  types  have  a 
multi-stage  air  compresser'  for  fuel- 
injection  and  starting  air,  driven  di- 
rectly by  a  crank  on  the  end  of  the  main 
shaft. 

The  speed  of  the  Type  A  engines 
ranges  from  135  to  170  r.p.m.;  and  of  the 
Type  B,  from  120  to  280  r.p.m. 

A  test  was  published  of  a  Type  A 
engine  having  a  cylinder  diameter  of 
18%  in.  and  a  stroke  of  28%  in.,  and  designed  to  develop  500  b.h.p.  at 
164  r.p.m.  Fig.  38  shows  an  indicator  diagram  taken  from  this  engine, 
and  Fig.  39  a  curve  of  fuel  consumption. 

The  Busch-Sulzer  Bros.  Diesel  engine  is  a  typical  vertical  Diesel  engine 


Lb.  Fuel  per  B.H.P.-HI 

O  ro  i»  O>  c 

> 

s^ 

• 

^ 

*—  *^. 

—   — 

0     ?0     40     60     80     100    120    140  160 
B.H.P. 

FIG.  37. — Fuel  consumption  of  a  De 
La  Vergne  type  "FH"  engine. 


46 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


and  is  built  by  the  Busch-Sulzer  Bros.  Diesel  Engine  Co.  of  St.  Louis. 
One  working  cylinder  and  the  two-stage  air  compressor  are  shown  in  sec- 
tion in  Fig.  40,  and  an  end  section  through  the  cylinder  in  Fig.  41. 

These  drawings  are  to  scale  and  a  good  idea  of  general  proportions  may 
be  obtained  from  them. 

Continental  Motors  are  built  by  the  Continental  Motors  Corporation, 
Detroit,  Mich.  This  company,  which  makes  motors  only,  has  several 
types,  suitable  for  automobiles  and  trucks.  They  are  all  water-cooled 
4-cycle  engines. 

Model  7W  motor  has  a  bore  of  3  34  in.  and  a  stroke  of  4^  in.  Fig.  42 
gives  a  capacity  curve  at  different  speeds,  showing  that  the  maximum  of 
42  b.h.p.  is  at  2140  r.p.m. 


R.P.M./6S 


FIG.  38. — Indicator  diagram 
from  a  Mclntosh  &  Seymour 
Diesel  engine. 


_b.  Fuel  per  B.H.P.-HF 

O  fvj  4*  0>  o 

X 

^ 

100 


200     300     400 
B.H.P. 


500     600 


FIG.  39.  —  Fuel  consumption  of  Mc- 
lntosh &  Seymour  Diesel  engine. 


Some  of  the  most  interesting  data  for  this  engine  are  given  below: 


Ratio  of  clearance  to  total  cylinder  volume  .......................   0  .  225 

Firing  order  .......  .'.....  ...............................    1-5-3-6-2-4 

Total  weight  ........  <  ..........................  ,  ............   575.  0  Ib. 

Flywheel,  16  in.  diameter  .........  '....'  ........................   62.  0  Ib. 

Compression  pressure-gage  ..................................     61  .  5  Ib. 

Crank    shaft:  3  bearing  —  2^   in.    diam.     Bushings    bronze,   babbitt  lined.     Front 

bearing  2%  2  by  2%e  in..     Center  bearing  2  Y±  by  3  in.     Rear  bearing  2J^2  by 

3^2  in. 
Cam  shaft:  3  bearing  —  lin.  diam.     Front  bearings  1%  by  2*^2  in.     Center  bearing 

1  l  K  e  by  2%  in.     Rear  bearing  1  l  K  6  by  2  >£  in. 
Connecting  rod  8%  (3.78  cranks)  long.     Crank-end  bearing  12%2  by  2  in.     Piston 

pin  bearing  %  by  1^  in. 
Piston  3)^  in.  long.     Three  rings,  %6  in.  wide. 

Model  E7  is  a  four-cylinder  truck  engine  with  two  cylinders  cast 
en  bloc  and  an  aluminum  crank  case.  The  cylinders  are  4%  in.  diameter 
with  a  S^-in.  stroke. 

Fig.  43  is  a  capacity  curve  for  a  Model  E7  engine. 


TILE  INTERNAL-COMBUSTION  ENGINE 


47 


Fio.  40.— Busch-Sulzer  Bros.  Diesel  engine. 


48  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIQ.  41. — Busch-Sulzer  Bros.  Diesel  Engine. 


THE  INTERNAL-COMBUSTION  ENGINE 


49 


The  maximum  power  is  44  b.h.p.  at  1320  r.p.m. 
Some  of  the  data  for  Model  E7  is  as  follows : 

Ratio  of  clearance  volume  to  total  cylinder  volume 0 . 239 

Compression  pressure-gage 55  Ib. 

Firing  order 1-3-4-2 

Total  weight 660  Ib. 

Flywheel,  17M  in-  diam.,  weight 105  Ib. 


40 


30 


20 


10 


400  800  1200  1600  2000  2400 

R.P.M. 

FIG.  42. — Power  curve  for  model  7W  Continental  engine. 


9V 

40 

0.' 

a:  30 

CO 

ZO 
I0> 

^ 

+  

~» 

pN 

/ 

/ 

/ 

> 

/ 

/ 

/ 

/ 

/ 

400            600              800              JOOO            '1200              1400           I60G 
R.P.M. 

FIG.  43. — Power  curve  for  model  E7  Continental  engine. 

Crank  shaft — 3  bearing — 2*4  in.  diam.     Bushings  bronze,  babbitted.     Front  bearing 

2K  6  by  3  in.     Center  bearing  2%  2  by  3%  in.     Rear  bearing  2  y±  by  41  %  6  in. 
Cam  shaft— 3  bearing— 1%  in.  diam.     Front  bearing  2>£  by  2£{6  in.     Center  bearing 
in.     Rear  bearing  2^  by  1  %  in. 


50  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Connecting  rod  11  in.  (4  cranks)  long.     Crank  end  bearing  2}£  by  3  in.     Piston  end 

bearing  1  J{6  by  2^  in. 
Piston,  5^  long.     Three  rings,  KG  wide. 

References 

The  Eight-cylinder  Automobile  Engine. — The  Gas  Engine,  March  and  April,  1915. 

Aeroplane  Motors  Used  by  the  United  States  Naval  Aeronautic  Corps. — The 
Gas  Engine,  Aug.,  1915. 

Diesel  Engines  Applied  to  Marine  Purposes. — The  Gas  Engine,  Jan.,  1916. 

Twelve-cylinder  Engines. — The  Gas  Engine,  Jan.,  1916. 

The  Problem  of  Airplane  Engines. — The  Gas  Engine,  June,  1917. 

The  Design  of  Motor  Truck  Engines  for  Long  Life. — The  Gas  Engine,  June,   1917. 

Practical  Operation  of  Gas  Engines  Using  Blast-furnace  Gas  as  Fuel. — Trans. 
A.S.M.E.,vol.  35,  p.  151. 

Heavy  Oil  Engines.— Jour.  A.S.M.E.,  Oct.,  1916. 

An  American  Racing  Engine. — The  Gas  Engine,  Nov.,  1916. 


PART  II-THERMQDYNAMICS 


CHAPTER  VI 
GENERAL  POWER  FORMULAS  AND  GASES 

23.  Power  of  Heat  Engines. — Heat-engine  cycles  are  usually  complete 
on  one  side  of  the  piston,  and  if  the  pressures  acting  on  that  side  as  the 
piston  travels  back  and  forth  are  plotted,  a  diagram  is  produced  which  is 
a  measure  of  the  work  done  during  the  cycle.  This  diagram  is  traced 
for  actual  engines  by  an  instrument  called  an  indicator,  and  all  such  dia- 
grams, either  actual  or  for  theoretical  cycles  such  as  the  Carnot  cycle, 
are  called  indicator  diagrams,  or  perhaps  more  commonly,  indicator 
cards. 

If  the  cylinder  is  double-acting,  two  cycles,  on  opposite  sides  of  the 
piston,  are  accomplished  at  the  same  time,  although  corresponding  strokes 
are  not  simultaneous. 

Should  the  cycle  depend,  for  the  performance  of  different  functions, 
upon  the  two  sides  of  the  piston,  or  upon  a  main  and  auxiliary  piston 
(as  in  2-cycle  gas  engines),  indicator  diagrams  must  be  obtained  from 
these  separately  and  the  net  work  computed.  However,  in  such  a  case 
the  real  work  of  the  cycle  is  done  on  one  side  of  the  piston  and  it  is  this 
side  which  will  be  considered  as  the  working  cylinder-end. 

In  practice  the  indicator  diagram  is  mostly  used  to  determine  the 
mean  effective  pressure  (m.e.p.),  which,  when  engine  dimensions,  speed  and 
the  cycle  on  which  the  engine  operates  are  known,  may  be  used  for  the 
determination  of  power,  or  conversely,  the  determination  of  cylinder 
dimensions  necessary  for  a  given  power. 

In  all  practical  engines,  all  strokes  are  equal,  so  let  v8  denote  the 
volume  swept  through  by  the  piston  during  a  complete  single  stroke,  in 
cu.  ft.  Let  pi,  pz,  Pz  and  p±  be  mean  pressures  in  Ib.  per  sq.  ft.  acting 
during  four  consecutive  strokes  of  a  4-stroke-cycle  engine,  on  one  side  of 
the  piston.  On  all  strokes  toward  the  cylinder-end  considered,  the  work- 
ing substance  resists  the  movement  of  the  piston  and  the  work  is  negative, 

The  pressure  in  the  working  cylinder-end  may  be  considered  as  abso- 
lute. The  pressure  on  the  opposite  side  of  the  piston  may  be  ignored 
as  it  does  not  affect  the  indicated  work  of  the  cycle, 

51 


52  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  work  of  the  cycle,  in  ft.  lb.,  for  one  cylinder-end  is: 
W  =  pivs  -  p2va  +  p3vs  -  p4vs  =  (pi  —  pz  +  PS-  Pt)vs  =  144PMvs     (1) 
where  PM  is  the  m.e.p.  in  lb.  per  sq.  in.,  of  the  whole  cycle,  referred  to 
one  stroke.     Any  other  cycle  would  give  the  same  result. 

Then: 


This  m.e.p.,  exerted  throughout  one  stroke  of  the  piston,  would  do 
the  work  of  the  cycle  for  the  working  substance  on  one  side  of  the  piston. 

The  indicator  diagram  is  drawn  with  the  length  representing  the 
length  of  stroke;  then  the  area  divided  by  the  length  gives  the  m.e.p. 
when  dimensions  are  in  terms  of  pressure  and  volume,  and  in  (2),  TF/144 
indicates  that  pressure  is  in  lb.  per  sq.  in. 

The  work  in  terms  of  heat  units  is: 

w  =  &  -  0« 

A 

where  Qi  and  Q2  are  heat  quantities  received  and  rejected  per  cycle, 
respectively,  and  A  is  the  reciprocal  of  Joule's  equivalent  (= 
Substituting  this  in  (2)  gives: 


vs  vs 

=  54  pB-t.u.  converted  into  work  perl  /^\ 

Leu.  ft.  of  piston  displacement  J 

Formulas  (2)  and  (3)  are  general  expressions  of  m.e.p.  for  all  heat 
engine  cycles,  theoretical  or  practical. 

Power  is  work  done  in  a  unit  of  time;  then  as  33,000  ft.  lb.  per  minute 
is  the  unit  of  horsepower,  the  work  done  per  minute  divided  by  33,000 
gives  the  horsepower.  Power  determined  by  means  of  an  indicator  dia- 
gram, which  is  the  power  developed  in  the  engine  cylinder,  is  called  indi- 
cated horsepower  (i.h.p.). 

Let  Nc  =  the  number  of  cycles  per  minute;  then: 

ihp    =J^£    =144P.jWVe  () 

33,000  33,000 

With  the  usual  slider-crank  mechanism  employed  for  reciprocating 
engines  there  are  two  strokes  per  revolution;  then  the  number  of  strokes 
per  min.  is  2N,  where  N  is  the  number  of  revolutions  of  the  crank  per 
minute  (r.p.m.).  If  m  denotes  the  number  of  strokes  per  cycle,  then  the 
number  of  cycles  per  min.  per  cylinder-end  is: 

AT          2N 
NC  =  - 

m 


GENERAL  POWER  FORMULAS  AND  GASES 


53 


Then  for  one  working  cylinder-end  (4)  becomes: 

2  X  144  PMvsN 


i.h.p. 


33,000         m 


(5) 


The  maximum  difference  in  pressure  on  tne  two  sides  of  a  piston  during 
a  cycle  is  called  the  maximum  unbalanced  pressure,  and  it  is  this  which 
determines  the  required  strength  and  consequent  weight  of  the  working 
parts  of  the  engine.  If  this  pressure  is  high  relative  to  the  m.e.p.,  the 
engine  will  be  heavy  for  the  work  done,  and  the  friction  may  more  than 
offset  the  advantage  of  an  economical  heat  cycle  if  this  is  carried  too  far, 
as  will  be  presently  shown. 

Brake  Horsepower  (b.h.p.)  is  the  power  delivered  at  the  engine  wheel, 
or  net  power,  as  measured  by  some  form  of  friction  brake  or  dynamometer. 

It  is  the  difference  between  the 
i.h.p.    and   the  friction  horsepower 

(f.h.P;). 

Figure  44  is  a  diagram  of  a 
simple  form  of  friction  brake.  The 
band  consists  of  blocks  of  wood 
held  together  by  band  iron,  with 
adjustment  at  (F).  The  engine 
wheel  rim  sliding  inside  this  band 
is  resisted  by  friction.  The  product 
of  this  resistance  in  Ib.  and  the  distance  traveled  by  the  rim  in  ft.  per 
min.  is  the  work  in  ft.  Ib.  per  min.  absorbed  by  the  brake.  This 
divided  by  33,000  gives  the  b.h.p. 

It  is  usually  inconvenient  to  measure  the  resistance  right  at  the  wheel 
rim,  so  a  brake  arm  of  length  I  is  made  long  enough  to  rest  upon  a  pedes- 
tal on  the  scale  platform.  If  P  is  the  effective  force  in  Ib.  after  deducting 
the  weight  of  pedestal  and  unbalanced  weight  of  brake  arm,  I  is  the  length 
of  the  arm  in  ft.  and  N  the  r.p.m.,  then: 


FlG   44 


b.h.p.  = 


2irlNP        INP 
33,000   =~  5252 


(6) 


The  b.h.p.  measured  in  this  way  may  be  applied  by  belting  to  machinery, 
by  direct-connecting  to  electric  generators  or  other  machinery,  or  thiough 
gears,  and  is  probably  not  an  exact  measure  of  the  net  work,  as  bearing 
friction  must  be  affected  differently  in  the  various  cases  of  -application. 

For  electrical  machinery  the  output  is  measured  at  the  switchboard  in 
kilowatts,  which  may  be  reduced  to  electrical  horsepower  (e.h.p.);  or, 

e.h.p.  =  1.34  X  kw. 


54  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

This  is  the  net  power  after  deducting  the  friction  of  engine  and  generator. 
If  the  friction  of  the  generator  is  known  the  b.h.p.  may  be  found. 

In  pumping  engines  pump  horsepower  is  the  indicated  work  done  in  the 
pump  cylinder,  the  difference  between  this  and  the  i.h.p.  being  the  f.h.p. 
Mechanical  efficiency  is  given  by  the  ratio: 

_  i.h.p.  —  f.h.p.  _  b.h.p.  (  . 

*M  =          i.h.p.          =  Lh^l 

The  friction  of  an  engine  does  not  vary  greatly  for  varying  loads,  so 
that  if  the  i.h.p.  is  small  relative  to  the  size  of  the  engine,  eM  is  small. 
Then  eM  decreases  as  i.h.p.  (which  is  directly  proportional  to  PM  at 
constant  speed)  decreases,  the  limit  being  zero — when  the  i.h.p.  is  just 
sufficient  to  run  the  engine  (see  Chap.  X). 

A  diagram  of  the  Carnot  cycle  for 
air  is  plotted  to  scale  in  Fig.  45. 
The  m.e.p.  of  such  a  diagram  is  very 
small  when  compared  with  the  max- 
imum pressure,  and  even  though 
Carnot's  cycle  were  practically  pos- 
sible, an  engine  employing  it  would 
probably  be  able  to  do  little  more  than 
overcome  its  own  friction  if  gas  were 
the  working  substance.  This  would 
FIG.  45.  n°t  be  true  of  steam,  the  m.e.p.  in 

one  case  being  about  four  times  that 

for  air  when  the  same  pressure  and  temperature  limits  were  assumed. 
Neglecting  friction,  if  a  cylinder  20  in.  in  diameter  were  used  on  a  steam 
engine,  the  air  engine  with  the  same  mean  piston  speed,  developing  the 
same  power  would  require  a  cylinder  about  40  in.  in  diameter. 

It  is  obvious  therefore  that  the  utility  of  Carnot's  cycle  for  gas  is 
limited  to  what  it  contributes  to  the  science  of  thermodynamics. 

24.  Practical  Cycles  for  Engines  Using  Gas. — In  practical  efficient 
engines  using  gas  (air  and  fuel  gas  or  vapor)  as  a  working  substance,  the 
combustion  of  fuel  takes  place  in  the  engine  cylinder.  This  necessitates 
a  fresh  supply  of  air  for  every  cycle,  with  a  consequent  exhausting  of  the 
burnt  gases.  This  may  be  accomplished  with  a  2-stroke  cycle  by  the  use 
of  auxiliary  cylinders;  or  suction  and  exhaust  strokes  may  be  added,  mak- 
ing a  4-stroke  cycle. 

The  constant-volume  and  constant-pressure  cycles  are  the  two  cycles 
now  employed  for  internal-combustion  engines,  and  while  the  theoretical 
efficiencies  are  greatly  in  excess  of  those  actually  attainable,  due  to 
the  impracticability  of  utilizing  such  high  temperature,  the  practical 


GENERAL  POWER  FORMULAS  AND  GASES 


55 


results  are  satisfactory  when  compared  with  the  use  of  steam  as  a  working 
substance.  Both  of  these  cycles  were  originally  intended  for  4-stroke 
cycles,  but  have  been  modified  to  operate  with  two  strokes.  In  their 
theoretical  consideration  this  makes  no  difference,  and  heat  transfer  to 
and  from  a  given  weight  of  gas  may  be  assumed. 

25.  The  Constant-volume  Cycle. — Figure  46  is  the  indicator  diagram 
for  the  theoretical,  4-stroke,  constant-volume  cycle,  commonly  called  the 
Otto  cycle.  Let  it  be  assumed  that  the  clearance  space  v\  contains  1  Ib.  of 
air  at  atmospheric  pressure.  In  practice  this  is  usually  a  mixture  of  air  and 
burnt  gas,  mostly  the  latter  unless  some  special  device  is  employed  for 
clearing  out  all  the  products  of  combustion,  which  is  known  as  scavenging. 

While  there  are  but  four  piston  strokes  per  cycle,  there  are  six  dis- 
tinct steps  as  follows: 

1.  Starting  from  the  position  shown  by  the  full  lines  in  Fig.  46,  with  the 
inlet  valve  open  and  exhaust  valve  closed  as  in  Fig.  29-A,  Chap.  V,  the  piston 
moves  through  the  distance  vs  representing  the  volume  of  stroke,  to  the 
position  given  by  the  dotted  lines  (Fig.  46).  This  is  the  suction  stroke, 
during  which  the  charge  of  fuel  and  air  (necessary  for  its  combustion)  is 
taken  into  the  cylinder.  In  some  engines  using  the  heavy  oils  with  the 
constant-volume  cycle,  only  air  is 
admitted  to  the  cylinder  proper 
during  the  suction  stroke,  the  fuel 
being  admitted  into  a  hot  bulb 
forming  part  of  the  combustion 
chamber,  or  sprayed  against  a  hot 
plate  or  into  a  hot  bulb  at  the  end 
of  the  compression  stroke.  With 
medium  compression  (280  to  300 
Ib.  per  sq.  in.  gage),  the  latter 
are  known  as  semi-Diesel  engines. 

The  suction  stroke  is  simply 
a  mechanical  step  and  plays  no 
direct  part  in  the  thermodynamics 


FIG.  46. — Constant-volume  cycle. 


of  the  cycle.  It  affects  the  power  by  using  time  that  might  have  been 
employed  for  work.  The  pressure  during  the  suction  stroke  is  less  than 
that  of  the  atmosphere  and  any  loss  resulting  is  charged  to  engine 
friction.  This  pressure  difference  will  be  neglected  in  the  discussion  of 
the  theoretical  cycle. 

2.  The  charge  is  next  compressed  adiabatically  along  line  2-3  during 
the  compression  stroke  while  both  inlet  and  exhaust  valves  are  closed,  as 
in  Fig.  29-B,  Chap.  V. 


56  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

3.  Ignition  is  now  effected,  usually  by  an  electric  spark  except  in  oil 
engines  using  the  hot  bulb  or  plate.     The  addition  of  heat  at  constant 
volume  along  line  3-4  is  assumed,  with  rise  in  temperature  and  pressure. 
The  heat  added  is: 

Q!  =  cv(T*  -  T$  (8) 

in  which  T  is  absolute  temperature  in  degrees  F.,  and  cv  the  specific  heat 
at  constant  volume. 

4.  Adiabatic  expansion  occurs  during  the  working  stroke  along  line 
4-5,  both  valves  remaining  closed  as  in  Fig.  29-C,  Chap.  V. 

5.  In  practical  engines  the  exhaust  valve  opens  at  or  near  the  end  of 
the  expansion  stroke,  and  the  drop  in  pressure  along  line  5-2  is  accompa- 
nied by  expansion,  the  temperature  T%  being  indefinite.     In  the  theoret- 
ical cycle  it  is  assumed  that  the  valves  remain  closed  and  that  heat  is 
withdrawn  along  line  5-2,  the  resulting  pressure  and  temperature  being  the 
same  as  at  the  end  of  the  suction  stroke.     This  completes  the  thermal 
cycle.     The  heat  rejected  during  the  step  is: 

Q,=c.(r,-r,)  0) 

6.  The  exhaust  stroke  2-1  is  one  of  the  two  preparatory  steps  for  the 
next  thermal  cycle,  removing  the  burnt  gas.     The  exhausjb  valve  is  open 
during  this  stroke  as  in  Fig.  29-D,  Chap.  V,  and  the  pressure  in  actual 
engines  is  slightly  above  atmospheric.     The  loss  is  charged  to  engine 
friction  as   with   the   suction    stroke,   and   the   pressure   difference   is 
neglected  for  the  theoretical  cycle.     This  completes  the  practical  cycle. 

The  efficiency  of  the  constant-volume  cycle  is  : 

_  Qi-Qa  _  cv(T,  -  T,)  -  cv(T,  -  T,)  T^-  T2 

~ 


Taking  exponent  n  =  k  for  adiabatic  changes  (in  which  k  =  cp/cv, 
cp  being  specific  heat  at  constant  pressure)  : 


or, 

T5      T*       T,  -  T 


T,      T,      T,  -  T 
Then  letting  r  —  Vz/vi  and  remembering  that: 


(10)  may  be  written: 

e 


GENERAL  POWER  FORMULAS  AND  GASES 


57 


FIG.  47. 


It  is  apparent  from  (11)  that  efficiency  is  improved  by  increasing  the 
compression  pressure,  which  is  true  in  practice  with  certain  limitations 
(see  Chap.  IX). 

The  expansion  and  compression  lines  of  actual  indicator  diagrams  are 
not  adiabatic,  but  are  approximated  by  the  equation: 

pvn  =  constant; 

where  n  is  some  value  usually  less  than  k. 
However,  with  any  value  other  than 
k  ( =  cp/cv),  Equation  (11)  does  not  apply; 
because  the  values  of  n  which  are  found 
approximately  from  actual  diagrams  are 
greater  or  less  than  k,  indicating  that 
heat  is  added  or  subtracted  as  the  work- 
ing substance  changes  volume.  These  quantities  of  heat  must  be  added  to 
or  subtracted  from  Qi  and  Q2  in  (10).  This  results  in  a  clumsy  formula 
with  assumed  values  of  n  at  best,  and  it  is  better  to  use  (11)  for  compari- 
son with  actual  efficiencies.  Practical  values  of  n  must  be  used  in  deter- 
mining the  clearance  volume  of  actual  engine  cylinders. 

Constant-volume  heat  transfer  involves  instantaneous  combustion  and 
exhaust;  this  is  impossible  in  practice,   causing  the  explosion  line  to 
lean.     This,  together  with  the  fact  that  the  valves  do 
not  open  and  close  exactly  at  the  ends  of  the  stroke, 
causes  rounded  corners  on  the  diagram. 

The  theoretical  maximum  temperature  is  not  per- 
missible due  to  the  burning  and  warping  of  the  metal, 
and  the  impossibility  of  lubrication,  so  approximately 
30  per  cent,  of  the  heat  of  combustion  must  be  removed 
by  the  circulation  of  water  around  the  cylinder.  This 
lowers  the  pressure  and  reduces  the  area  of  the  diagram. 
An  actual  indicator  diagram  from  an  engine  operating  on 
the  4-stroke  Otto  cycle  is  shown  in  Fig.  47. 

A  critical  examination  of  the  Otto  cycle  is  given  in 
Giildner's  Internal-combustion  Engines. 

The  T<J>  (temperature-entropy)  diagram  for  the  Otto 
cycle  is  shown  in  Fig.  48,  the  numerals  corresponding  to 
the  same  points  on  the  pv  diagram.     The  reception  of  heat 
at  constant  volume  is  along  the  line  3-4  and  adiabatic  expansion  along  4-5. 
Rejection  of  heat  at  constant  volume  is  along  line  5-2  and  adiabatic  com- 
pression along  2-3. 

From  general  thermodynamics,  line  3-4  is  drawn  by  means  of  the 
equation: 


FIG.  48. 


58  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Tx 

A0    =    Cv    \0ge  TfT 
i  3 


and  line  2-5  from: 

A<£  =  cv  logc 


T< 


where  Tx  is  any  temperature  above  jP3  or  TV 

26.  The  Constant -pressure  Cycle. — The  Diesel  cycle  is  the  best  known 
and  most  widely  used  of  this  class.  The  theoretical  difference  between 
this  cycle  and  the  Otto  cycle  is  the  combustion  of  fuel  at  constant  pres- 
sure instead  of  at  constant  volume.  Diesel's  original  purpose  was  to 

approach  the  Carnot  cycle  by  applying  isother- 
mal combustion,  but  practical  results  with  the 
later  engines  employing  constant-pressure 
combustion  show  improvement  over  the  former 
method,  and  the  cycle  is  now  generally  known 
as  the  constant-pressure  cycle. 

The  theoretical  indicator  diagram  for  the 
Diesel   4-stroke    cycle   is  shown  in  Fig.  49. 
FIG.   49. -constant-pressure    The  suctiOn  and  exhaust  strokes,  like  those 

of  the  Otto  cycle,  are  purely  mechanical,  and 

the  operation  of -the  inlet  and  exhaust  valves  are  the  same  as  for  the 
Otto  cycle. 

Some  form  of  liquid  fuel,  like  crude  petroleum,  has  thus  far  been  the 
only  fuel  used  in  the  Diesel  engine. 
The  six  steps  of  the  cycle  are: 

1.  The  suction  stroke,  line  1-2,  during  which  fresh  air  only  is  drawn 
into  the  cylinder. 

2.  The    compression    stroke,    along    line    2-3,    compresses    the  air 
adiabatically  to  a  pressure  of  about  500  Ib.  per  sq.  in.  gage,  and  a  tem- 
perature high  enough  to  positively  ignite  the  fuel,  although  none  has  been 
supplied  up  to  this  time. 

3.  The  fuel  valve  is  now  opened  (slightly  before  the  end  of  the  stroke), 
and  as  the  piston  starts  upon  its  stroke,  oil  is  injected  in  the  form  of 
spray  at  a  rate  which  allows  the  pressure  to  remain  practically  constant 
as  the  fuel  burns  along  line  3-^.     Ignition  is  effected  by  the  heat  of  com- 
pression and  no  electric  spark  is  required.     The  amount  of  fuel  in- 
jected is  controlled  by  the  governor  and  depends  upon  the  load  on  the 
engine. 

The  air  used  for  fuel  injection  is  furnished  at  a  pressure  from  50  to 
100  per  cent,  in  excess  of  the  engine  compression  pressure,  by  either  an 
independent  compressor  or  by  a  compressor  connected  to  the  engine. 


GENERAL  POWER  FORMULAS  AND  GASES  59 

Sometimes  the  fuel  is  pumped  directly  into  the  cylinder  without  the  use 
of  compressed  air. 

The  heat  received  is: 

Ql  =  cp(Tt  -  Tj  (12) 

4.  The  remainder  of  the  working  stroke  is  occupied  with  adiabatic 
expansion  along  line  4-5. 

5.  The  rejection  of  heat  is  assumed  at  constant  volume  along  line 
5-2,  completing  the  thermal  cycle. 

The  heat  rejected  is: 

Q2  =  cv(T,  -  T2)  (13) 

6.  The  exhaust  stroke  removes  the  burned  charge  from  the  cylinder, 
completing  the  practical  cycle. 

The  efficiency  of  the  Diesel  cycle  is: 

=  Qi-<?2  =  cp(T,  -  T3)  -  cv(T,  -  T2)  _  cv(T5  -  r2) 

Q1  cP(T,  -  Tt)  ~ 


=5fj  (14) 

This  may  be  written: 

m     ( •*•  5 

1  2  l-^T    — 


J 


But: 

\Z>2 

And  from  Charles'  law, 
Then: 

rji  fjj     *    rji 

J.  2  J.  4        J.  3 

Substituting  in  (15)  and  letting  vz/vi  =  r,  as  for  the  constant-volume 
cycle,  and  v*/v\  =  e,  (15)  becomes: 

e=l-  ^  ~  ^  (16) 

This  may  be  written: 


If  the  quantity  in  brackets  is  omitted  (17)  is  the  same  as  (11),  and 
as  this  quantity  is  always  greater  than  unity  it  is  obvious  that  for  the 


60 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


same  value  of  r,  the  efficiency  of  the  constant-volume  cycle  is  greater 
than  that  of  the  Diesel.  The  greater  value  of  r  in  the  Diesel  cycle,  allow- 
able because  of  the  absence  of  fuel  during  compression,  accounts  for  the 
superior  economy  usually  attained  by  the  Diesel  engine. 

An  actual  indicator  diagram  from  a  Diesel  engine  is  shown  in  Fig.  50. 

Theoretical  and  actual  efficiencies  may  be  obtained  for  both  the  con- 
stant-volume and  constant-pressure  cycles  in  terms  of  heat  quantities 
based  upon  fuel  supplied,  in  which  case: 

e  =  _  2545 

B.t.u.  per  horsepower  per  hr. 

The  T<f>  diagram  for  the  Diesel  cycle  is  shown  in  Fig. 
51.  It  is  similar  to  Fig.  48  except  that  the  reception 
of  heat  is  at  constant  pressure  along  line  3-4,  the  equation 
being: 

A0  =  c 

The  rejection  of  heat 
along  line  5-2  is  at  con- 
stant volume. 

As  Tz/Ts  is  not  equal 
to  T5/Tt,  the  relation  of 
the  efficiency  formula  to 
the  diagram  is  not  ap- 
parent, as  it  is  with  the  constant-volume  cycle. 

27.  Volumetric  Efficiency.  —  The  volumetric  efficiency  of  an  internal- 
combustion  engine  is  the  ratio  of  the  volume  of  the  charge  (fresh  air  and 
fuel)  vc  at  some  standard  pressure  and  temperature,  to  the  volume  of 
stroke  va.  This  is  equivalent  to  the  ratio  of  the  actual  charge  weight 
wc  to  the  weight,  at  some  standard  pressure  and  temperature,  of  a  volume 
of  the  same  substance  equal  to  v,,  the  volume  of  stroke;  let  this  weight 
be  wa.  The  weight  wcj  at  actual  pressure  and  temperature  will  occupy  the 
volume  qvs,  where  q  is  a  factor  depending  upon  the  relative  volume  of 
fresh  charge  to  total  cylinder  contents.  In  theoretical  computations  it 
is  usually  assumed  that  q  is  unity.  With  poor  valve  design  and  setting 
q  may  be  less  than  unity,  and  with  exceptionally  good  valve  design  and 
setting  it  may  be  greater.  ,  With  complete  scavenging,  when  all  the  burnt 
gas  is  expelled  from  the  cylinder: 


FIG.  50. 


r  —  1 


GENERAL  POWER  FORMULAS  AND  GASES  61 

The  general  equation  for  gas  is: 

pv  =  wRT 

in  which  R  is  a  constant  (  =  53.35  for  air). 

Letting  the  subscript  2  refer  to  actual  cylinder  conditions,  and  absence 
of  subscripts  to  a  standard  condition,  we  may  then  write: 

Pz  Q^s        i  P^s 

Then: 


.  =     . 

vs      ws      *  pT2 

The  factor  q  is  difficult  of  practical  determination  but  is  of  considera- 
ble importance  in  connection  with  economy  and  capacity.  Aside  from 
affecting  the  volume  of  charge  it  affects  the  density  by  its  influence 
upon  TV,  and  any  measure  which  may  be  taken  to  increase  it  by  proper 
valve  setting  will  increase  both  economy  and  capacity. 

28.  Temperature  rise  due  to  combustion  must  be  known  in  order  to 

determine  theoretical  efficiencies  of  internal-combustion-engine  cycles. 

Let  h  =  the  heating  value  per  cu.  ft.  of  gaseous  fuel  or  per  Ib.  of  liq- 

uid fuel.     For  gas  this  must  be  at  some  standard  pressure 

and  temperature. 

a  =  cu.  ft.  of  air  supplied  per  cu.  ft.  of  gaseous  fuel  or  per  Ib.  of 

liquid  fuel  at  standard  pressure  and  temperature. 
<r  =  the  volume  of  fuel  for  which  a  is  supplied. 
c  =  specific  heat  in  general  —  cv  or  cp. 

Referring  to  Fig.  46,  Par.  25,  and  Fig.  49,  Par.  26,  letting  wz  be  the 
weight  of  total  cylinder  contents  in  Ib.,  the  B.t.u.  necessary  to  raise  the 
temperature  from  T$  to  TI  is: 

-  T7,). 


The  heat  supplied  per  cu.  ft.  of  mixture  at  standard  pressure  and 
temperature  is: 

h 
a  -+-  <7 

The  volume  occupied  by  the  charge  at  standard  pressure  and  tempera- 
ture is  : 

evv8 

where  ev  is  found  by  the  method  of  the  last  paragraph. 
Then  the  total  heat  supplied  per  cycle  is: 

»  =  e*  '• 


62  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Then: 

w2c(T,  -  T3)  =  ev  va  — ^— 
a  -f-  o" 

But,      .- 

and  ys  =  v2  -  VL 


Substituting  these  in  (20),  the  equation  may  be  written: 


— •   t  -~-   o  v  v  I 

a  +  < 
774  —  TZ  may  be  written: 

3^3  T—  -  1  I 
*  I  f  ' ' 

Substituting  in  (21)  gives: 


From  thermodynamics  : 

1*= 


r  -  1 


Substituting  in  (22)  gives: 

7%  =  /i         _fi_ 

T3  'v  a  +  d  '  c2  ' 


In  theoretical  cycles  it  is  common  to  assume  the  pressure  and  tempera- 
ture the  same  as  the  standard  at  which  a  and  h  are  measured,  and  that  the 
volume  of  charge  equals  the  volume  of  stroke;  but  as  this  is  in  no  wise  a 
necessary  assumption,  ev  will  be  retained  in  the  equations.  Its  value  is 
given  in  (19).  The  value  of  a  for  theoretical  cycles  is  usually,  but  not 
necessarily,  assumed  as  just  sufficient  to  support  perfect  combustion. 

The  quantity  <r  is  obviously  unity  for  gaseous  fuel.  For  oil  engines 
it  is  the  volume  of  1  Ib.  of  oil  vapor,  and  as  this  is  about  3  per  cent,  of  the 
volume  of  air  supplied,  it  is  usually  ignored. 

Standard  pressure  is  taken  as  14.7  Ib.  per  sq.  in.  absolute.  Standard 
temperature  is  usually  32  degrees  F.,  but  sometimes  62  degrees.  The 
A.S.M.E.  has  adopted  60  degrees  F. 

Taking  numerical  values  and  making  substitution,  assuming  the 
mixture  to  have  the  same  characteristics  as  air,  special  equations  may  be 
written. 

For  the  constant-  volume  cycle  using  gas,  c  =  cv,  and: 

p_4    r4=       316.        h 

PS      T9  PS  a  -\-  1 


GENERAL  POWER  FORMULAS  AND  GASES 


63 


For  constant-volume  cycle  for  oil: 

»4      TI           .   316  h  f        .,, 

-  =  —  =  1  -| .  6v  .     (r  -  i 

PS       1  3  PS  & 

For  the  constant-pressure  cycle  (Diesel) : 

vi  .  225  hf 

C^  =  1+^'e»'a(r-1) 


(25) 


(26) 


For  values  of  h  and  a  see  Par.  75,  Chap.  XIV. 

29.  The  mean  effective  pressure  of  any  heat  engine  is  given  by  Equa- 
tion (3),  Par.  23,  and  is: 

PM  =  5Ae^  (27) 

where  Qi  is  the  heat  furnished  per  cycle  and  e  the  thermal  efficiency. 
Substituting  the  value  of  Qi  from  (20)  in  (27)  gives: 


PM  =  5Aeet 


(28) 


which  is  the  m.e.p.  in  Ib.  per  sq.  in.  for  any  internal-combustion  engine. 
The  quantities  a  and  a  may  be  as  in  the  preceding  paragraph. 
Then  for  all  gas  engines,  h  is  heating  value  per  cu.  ft.,  and: 

^  (29) 


PM  =  5Aeet 

a  -h  1 

For  all  oil  engines,  h  is  heating  value  per  Ib.,  and: 

PM=  5.466,,  •-       • 


a 


(30) 


For  theoretical  cycles  e  and  a  may  be  theoretical  values;  or  if  actual 
practical  values  are  used,  PM  will  be  the  actual  m.e.p.  For  values  of 
h  and  a  see  Par.  75,  Chap.  XIV. 

30.  Conventional  indicator 
diagrams,  drawn  to  scale  to 
give  a  certain  m.e.p.  are 
sometimes  convenient  in  the 
absence  of  actual  diagrams, 
for  calculations  concerning 
strength  or  crank  effort.  The 
m.e.p.  may  be  determined 
from  the  preceding  para-  FIG.  52. 

graph,   using  practical  values 

as  nearly  as  possible.  Then  a  value  of  n  may  be  chosen  for  the  exponent 
of  the  curve  equation  which  will  give  values  approximating  actual  ex- 
pansion and  compression  curves.  The  fundamental  equations  used 
relate  to  pressure  and  volume  only. 


64 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Constant-volume  Cycle. — -As  in  the  previous  discussion,  all  pressures 
except  PM  are  in  Ib.  per  sq.  ft.,  and  subscripts  correspond  to  points  on 
Fig.  52. 

(31) 
(32) 
(33) 


Vl 


clearance  = 
The  work  per  cycle  in  ft.  Ib.  is : 


r  -  1 


=    JW±_    (}     _  1_\  _         P3^1       A  1     \ 

n  -  I  \         r»-V       n  -  1  \         r"-1/ 


144PM 


(P4  -  PS) 


~  r—V 


—  vi  (n  —  l)(r  —  1) 

Tf  PM  is  known  the  pressure  rise  in  Ib.  per  sq.  in.  is: 
P4  -  p3      P^(n  -  l)(r  -  1) 


144 


(34) 
(35) 
(36) 


1  - 


FIG.  53. 


Constant -pressure  Cycle 
(Diesel). — The  values  of  r  and 
rn-1  and  the  clearance  are  given 
by  (31),  (32)  and  (33)  as  for  the 
constant-volume  cycle. 

A  convenient  ratio  is: 


t/4 
€    =   — 

(37) 

Ratio  of  expansion  =  —  =  - 

(38) 

Out  off        V*  ~  Vl       €  ~  1 

(39) 

1)%  —  v\       r  —  1 

The  work 

per  cycle  in  ft.  Ib.  is: 

W  =  p-( 

/            \    ,     Psv*    f"1         /V4\n~1~\         p&i     /1         1   \ 

(40) 

l\^4             PjJ        I                       '1     I    -L     "~      \          I                                                 i      I  •"•          "      nil 

n  —  1  L          \v<2.'       J       n  —  l  \        r     / 

1A 

w         P3   \          '-1-^1 

A  T3                                                            t^d                             t        1 

.A.  /-*  .  .     —                             —     -  ...    1  £.    —    1     —I—                                               1 

(41) 

r  -  1  L  n  -  1 

If  PM  is  known,  e  may  be  found  by  trial  and  error. 


CHAPTER  VII 
STEAM 

A  knowledge  of  the  fundamentals  of  thermodynamics  is  desirable,  but 
not  essential  for  a  practical  understanding  of  the  formulas  expressing 
the  properties  of  steam.  For  equations  involving  entropy,  such  know- 
ledge is  necessary  and  is  presupposed.  Introductory  to  Chap.  XV,  a 
brief  review  of  a  few  of  the  principal  equations  will  be  given  in  this 
chapter.  It  will  also  give  a  review  of  these  principles  for  general  use. 

31.  Formation  of  Steam  under  Constant  Pressure. — When  steam  is 
"raised"  in  a  boiler  which  is  initially  partly  filled  with  comparatively 
cold  water,  heat  is  added,  and  after  a  certain  temperature  is  reached  the 
water  boils  and  steam  is  given  off.  The  pressure  rises  gradually  until 
the  desired  pressure  is  reached.  If  steam  is  now  drawn  from  the  boiler 
for  the  operation  of  a  steam  engine,  more  steam  is  formed  and  the  pres- 
sure is  maintained  as  nearly  constant  as  possible.  Then  we  may  say 
that  all  the  steam  which  is  formed  after  the  engine  begins  to  work  is 
formed  under  constant  pressure;  that  is,  every  pound  of  water  which 
.enters  the  boiler  at  a  certain  temperature  is  heated,  and  converted  into 
steam  at  a  constant  pressure. 

Steam  is  formed  in  this  way  for  most  of  its  practical  applications,  and 
the  heat  quantities  in  the  steam  tables  are  based  on  this  assumption. 
The  study  of  the  properties  of  steam  is  also  simplified  by  assuming  its 
formation  at  constant  pressure. 

Let  a  vertical  cylinder  contain  1  Ib.  of  pure  water  upon  which  rests  a 
frictionless  piston,  exerting  a  constant  pressure  p  upon  the  water.  With 
the  water  in  the  cylinder  at  initial  temperature  to,  assuming  no  leakage 
of  water,  steam  or  heat,  the  three  stages  of  steam  formation  are  as 
follows: 

1.  Heat  is  added  and  the  temperature  gradually  rises  from  t0  to  some 
temperature  t  at  which  steam  begins  to  form.     This  temperature  depends 
upon  the  pressure  exerted  by  the  piston,  and  for  every  value  of  p  there 
is  a  certain  value  of  t,  at  which,  if  more  heat  be  added,  water  will  be  con- 
verted into  steam. 

2.  Heat  is  still  added,  and  the  water  at  the  temperature  corresponding 
to  the  pressure  p,  begins  to  form  into  steam  at  the  same  temperature  and 

s  65 


66  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

under  the  constant  pressure  p.  The  movement  of  the  piston  due  to  the 
expansion  of  the  water  as  it  is  heated  from  to  to  t  during  stage  (1)  is  so 
slight  as  to  be  negligible;  but  now  the  piston  rises  noticeably  due  to  in- 
crease of  volume  as  steam  is  formed,  and  continues  to  rise  until  all  the 
water  is  converted  into  steam.  The  steam  during  this  stage  is  saturated 
steanij  being  in  contact  with,  and  at  the  same  temperature  of  the  water 
from  which  it  is  formed.  At  the  end  of  the  stage,  when  all  the  water  is 
converted  into  steam  it  is  called  dry  saturated  steam  and  its  volume  is 
denoted  by  s. 

3.  As  more  heat  is  added  to  this  dry  saturated  steam  the  temperature 
rises  to  some  temperature  ts  and  there  is  -an  increase  of  volume  if  the  pres- 
sure remains  constant.  The  steam  is  now  superheated.  Each  degree 
rise  in  temperature  above  the  temperature  of  saturation  is  called  a  degree 
of  superheat. 

Had  the  volume  remained  constant  as  the  heat  was  added,  the  pres- 
sure would  have  risen.  This  is  called  superheating  at  constant  volume; 
but,  as  with  the  formation  of  saturated  steam,  constant  pressure  has  the 
most  practical  application,  especially  in  power  plant  operation,  and  is 
assumed  in  this  book. 

When  saturated,  the  characteristics  of  steam  differ  considerably  from 
those  of  a  gas,  but  approach  them  more  nearly  as  superheating  is  increased. 

Superheated  steam  at  a  given  pressure  may  have  any  practical  tem- 
perature higher  than  that  due  to  saturation  at  that  pressure. 

32.  The  relation  of  pressure  and  temperature  in  saturated  steam 
has  been  determined  by  direct  experiment,  the  results  being  given  in 
steam  tables.     A  pressure-temperature  curve  shows  that  the  pressure 
rises  with  the  temperature  at  a  rate  which  increases  rapidly  with  in- 
crease of  temperature. 

For  water,  as  is  the  case  when  feedwater  is  b.eing  pumped  into  a 
boiler,  the  pressure  may  be  greater  than  that  corresponding  to  its  tem- 
perature in  the  steam  tables,  but  never  less.  Should  it  be  made  less, 
some  of  the  heat  in  the  water  would  be  given  up  to  evaporate  a  portion 
of  the  water,  evaporation  continuing  until  the  temperature  was  reduced 
to  correspond  to  the  pressure.  The  heat  of  the  liquid  always  corresponds 
to  the  temperature  and  is  dependent  upon  it}  regardless  of  the  pressure. 

33.  Supply  of  Heat  in  the  Formation  of  Steam  at  Constant  Pressure.— 
The  arbitrary  zero  of  the  steam  tables  is  at  32  degrees  F.,  and  all  heat 
quantities  are  measured  from  this  zero.     The  heat  required  to  raise  the 
temperature  of  the  water  from  32  degrees  F.  to  that  of  the  boiling 
temperature  t  is  called  the  heat  of  the  liquid,  and  is  expressed  by: 

h  =  c(t  -  32)    •  (1) 


STEAM  67 

in  which  c  is  the  specific  heat,  which  is  nearly  unity  at  ordinary 
temperatures. 

Most  saturated  steam  is  not  entirely  dry,  the  fraction  x  of  a  given 
weight  being  steam,  the  remainder  water. 

The  heat  taken  in  as  water  is  changed  to  steam  at  constant  pressure 
is  called  the  latent  heat,  and  is  denoted  by  L.  The  total  heat  of  dry  satu- 
rated steam  is  given  by  the  formula: 

H  =  h  +  L  (2) 

Let  C  denote  the  heat  content  of  steam  in  any  condition  from  water 
to  superheated  steam.  Then  wet  steam' is  expressed  by: 

C  =  h  +  xL  .  (3) 

The  heat  content  of  superheated  steam  is: 

C  =  h  +  L  +  cP(ts  -  t) 

=  H  +  cP(ts  -  t)  (4) 

in  which  CP  is  the  specific  heat  at  constant  pressure. 

34.  Adiabatic    expansion   is  never  realized  in  practice,  but  is  ap- 
proached in  the  flow  of  steam  through  nozzles.     In  such  expansion  from  a 
pressure  indicated  by  subscript  1  to  a  lower  pressure  denoted  by  2,  this 
would  give : 

*'••  01    =    02  (5) 

in  which  0  is  the  entropy  of  the  steam.     The  entropy  of  wet   steam  is: 

xL 

where  T  is  the  absolute  temperature  in  degrees  F.  and  0  is  the  entropy  of 
water  above  32  degrees  F.  This  may  be  found  in  the  steam  tables,  as 

may  also  the  entropy  of  evaporation,  ~  •    If  #  is  known  for  one  pressure, 

it  may  be  found  for  the  other  by  combining  (5)  and  (6). 
The  entropy  of  superheated  steam  is : 

in  which  CP  is  the  mean  specific  heat  between  Ts,  the  temperature  of  the 
steam,  and  T7,  the  absolute  temperature  due  to  the  pressure. 

35.  The  Rankine  Cycle. — If  Ci  and  C2  are  heat  contents  at  the  same 
entropy,  it  may  be  shown  that  for  any  initial  and  final  condition  of  the 
steam  the  efficiency  of  the  Rankine  cycle  with  complete  expansion  is 
given  by: 

e  =  ^—^  (8) 

Gi   —   Il2 


68  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  mean  effective  pressure  for  the  cycle  in  Ib.  per  sq.  in.  is: 

PM  =5.4^1^  (9) 

Vc  —  ff 

where  vc  is  the  volume  after  expansion,  in  cu.  ft.,  of  1  Ib.  of  steam,  and 
a  is  the  volume  of  1  Ib.  of  water  (=  0.017,  nearly). 
For  the  Rankine  cycle  with  incomplete  expansion: 

r          r     ,    (Pc  - 

CI-GC+- 


where  Cc  is  the  heat  content  at  'the  same  entropy  at  terminal  pressure, 
P2  the  back  pressure  in  Ib.  per  sq.  in. 
The  mean  effective  pressure  is: 

PM  =  5.4  Cl~Cc  +  Pc.  -  P2  (11) 

vc  —  ff 

36.  Flow  of  Steam.  —  It  may  be  shown  that  the  frictionless  flow  of 
steam  in  ft.  per  sec.,  when  the  initial  velocity  is  zero,  is  expressed  by: 


-  C2  (12) 

If  pz  is  less  than  about  0.58pi,  it  is  found  by  experiment  that  the  jet  will 
not  retain  its  form  upon  leaving  the  nozzle,  and  a  diverging  nozzle 
must  be  used. 

37.  Effect  of  Friction  on  Steam  Flow. — If  the  fraction  y  of  the  heat 
drop  from  initial  pressure  p\  to  throat  or  exit  pressure  (either  of  which 
may  be  taken  as  p2)  is  required  to  overcome  the  surface  friction  of  the 
nozzle,  the  velocity  will  be: 


VN  =  U  -  y)(Ci  ~  C2)  (13) 


The  heat  quantity  y(C\  —  C2)  is  returned  to  the  steam,  increasing  the 
heat  content,  dry  ness  factor  and  entropy  at  pressure  p%.  If  XN  is  the 
new  dryness  factor  (x2  being  that  due  to  adiabatic  expansion  to  p2), 

hz  +  xNLz  =  hz  +  XzLz  +  y(Ci  -  C2) 
or, 

*  =  *2  +  «%=^  (14) 

^2 

and, 

•^    =    03+^  (15) 

The  specific  volume  is: 


=  <r 


(16) 


STEAM  69 


Where  u  is  the  change  of  volume  from  water  to  steam. 
The  area  of  opening  required  in  sq.  in. 


144a  =  (17) 

V  N 

With  high  initial  superheat,  steam  at  the  throat  of  a  nozzle  may  still 
retain  superheat.     Then  (14)  would  be  replaced  by: 

#2  +  CP(TN  -  T2)  =  #2  +  cP(Ts  -  T2)  +  y(d  -  C2) 
From  which: 

rw  =  T,  +  y(Cl  ~  Cz)  (is) 

Cp 


Then: 


=  e  +  -*  +  CP  log*  (19) 


TV  is  the  resulting  absolute  temperature,  Ts  the  temperature  due  to 
adiabatic  expansion,  and,  due  to  the  small  difference,  CP  has  been  assumed 
to  be  the  same  on  both  sides  of  the  original  equation.  Such  problems 
are  easily  solved  with  entropy  table  or  chart. 

A  barely  possible  case  would  be  the  change  from  slightly  wet  steam  to 
superheated  steam  ;  then  : 

A2  +  L2  +  cP(TN  -  T2)  =  h,  +  x2L2  +  y(d  -  C2) 
and 


Ts  _  T,  +  (2Q) 

Cp 

The  entropy  is  given  by  (19). 

The  energy  lost  by  friction  and  returned  to  the  steam  as  heat  may  also 
be  expressed  in  terms  of  velocity;  letting  q  =  VN/V,  the  ratio  of  actual 
to  theoretical  velocit  : 


Heat  returned  =      (F*  -  IV)  =  (21) 


This  may  replace  y(Ci  —  C2)  in  (14),  (18),  and  (20),  making  general 
equations  for  any  passages  when  velocity  decrease  due  to  friction  is 
known  or  assumed. 


CHAPTER  VIII 

ECONOMY 
Notation. 

P  =  pressure  in  Ib.  per  sq.  in. 
v  =  volume  of  steam  in  cu.  ft.  per  Ib. 
<7  =  volume  of  water  in  cu.  ft.  per  Ib. 
C  =  heat  content  in  B.t.u.  per  Ib.  above  32  degrees  F.,  of  steam  of 

any  condition. 

h  =  heat  in  the  water  in  B.t.u.  per  Ib.  above  32  degrees  F. 
w  =  actual  steam  consumption  per  horsepower  per  hour,  usually, 

but  not  necessarily  in  terms  of  i.h.p. 
WR  =  theoretical  steam  consumption  for  the  Rankine  cycle  per  horse- 

power-hour. 

e  —  actual  thermal  efficiency. 

eB  =  thermal  efficiency  at  brake,  or  economic  efficiency  =  eMe. 
eM  =  mechanical  efficiency  (see  Chap.  X). 
F  =  heat  factor,  or  ratio  of  actual  to  theoretical  efficiency. 
38.  The  Steam  Plant.  —  Formulas  generally  used  for  expressing  the 
thermal  efficiency  of  the  steam  engine  or  turbine  are: 

42.42  & 

a    —   _  (  1  ) 

B.t.u.  per  h.p.  per  min. 

e  = 


B.t.u.  per  h.p.  per  hr. 

Both  forms  are  used  by  different  writers,  the  second  being  that  given 
in  the  Rules  for  Conducting  Steam  Engine  Tests  by  the  A.S.M.E.,  with 
the  exception  that  the  constant  given  is  2546.5.  As  the  slide  rule  is 
usually  accurate  enough  for  such  work  the  simpler  and  more  usual  con- 
stant will  be  used  in  this  book. 

Though  (1)  and  (2)  are  simple  expressions,  they  permit  of  several 
interpretations.  If  the  object  of  a  test  is  to  determine  the  economy  of 
the  complete  power  plant,  the  heat  supplied  to  all  the  auxiliaries  must  be 
added  to  that  supplied  to  the  engine  or  turbine.  The  total  heat  supplied 
for  power  purposes  is  the  difference  between  the  heat  content  of  all  steam 
used  by  engines  and  auxiliaries,  the  pressure  and  quality  being  measured 
near  their  respective  throttle  valves;  and  the  heat  content  of  all  feed 

70 


ECONOMY  71 

water  measured  near  the  boilers,  provided  that  steam  for  heating  or 
industrial  processes  is  not  taken  from  the  same  boilers.  In  this  case, 
equivalent  water  at  raw  water  temperature  should  be  deducted  from  the 
total  feed  water. 

The  determinat  on  of  heat  quantities  and  the  apparatus  employed  are 
fully  discussed  in  treatises  on  mechanical  laboratory  methods  and  in  Rules 
for  Conducting  Steam  Engine  Tests,  Report  of  Power  Test  Committee, 
Trans.  A.S.M.E.,  vol.  37.  When  all  heat  quantities  are  obtained,  the 
efficiency  may  be  obtained  by  (2). 

39.  The  Prime  Mover. — For  the  engine  or  turbine  designer  the 
economy  of  the  engine,  independent  of  its  auxiliaries,  is  perhaps  of  the 
most  importance.  The  steam  consumed  by  the  engine  alone  is  then  con- 
sidered, and  as  the  condensed  exhaust  steam  is  available  for  feed-water 
whether  so  applied  or  not,  it  is  assumed  that  the  temperature  of  the  feed- 
water  before  heating  is  that  corresponding  to  the  back  pressure.  Then 
the  heat  delivered  to  the  engine  per  Ib.  of  steam  is: 

Ci  —  h2 

and  if  w  is  the  steam  consumption  (also  called  water  rate)  in  Ib.  per  horse- 
power-hour, the  B.t.u.  per  horsepower-hour  for  the  engine  will  be: 

w(d  -  h). 

Should  there  be  steam  jackets  or  a  reheating  receiver,  the  heat  supplied 
to  these  should  be  added,  as  they  directly  affect  the  cylinder  efficiency. 
As  the  condensed  steam  from  jacket  and  receiver  is  available  for  feed- 
water  at  the  pressure  of  supply,  which  is  usually  that  of  the  initial  cylinder 
pressure,  hz  =  hi.  If  Wi  is  the  weight  of  steam  supplied  to  jacket  or  re- 
ceiver, or  both,  the  heat  will  be: 

ti>i(Ci  -  hi). 

Then  the  total  B.t.u.  per  horsepower-hour  will  be: 

w(Ci  -  h*)  +  wi(Ci  -  hi). 
The  efficiency  will  be: 

= 2545 .  } 

~~  w(Ci  -  h2)  +  Wl(d  -  h^ 

As  steam  jackets  are  not  very  commonly  used,  and  their  use  is  decreas- 
ing with  the  use  of  superheated  steam,  they  may  ordinarily  be  neglected. 
Then  (3)  becomes: 

/  A\ 

"  w(C1  -  A,) 
The  water  rate  of  Rankine's  cycle  for  complete  expansion  is: 

2545  ,.. 

Wr  =      ^  W 


72  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

And  for  incomplete  expansion : 


-  c, 


5.4 


The  subscript  c  refers  to  terminal  pressure. 

Formula  (5)  is  useful  in  steam-turbine  design. 

40.  I.H.P.  or  B.H.P. — For  an  intelligent  conception  of  thermal 
efficiency  and  economy  the  term  horsepower  must  be  defined.  This  is 
commonly  .taken  as  i.h.p.  or  as  b.h.p.  Then  (2)  may  be  written: 

e  =  2545 

B.t.u.  per  i.h.p.-hr. 
or: 

2545 


B.t.u.  per  b.h.p.-hr. 

Formula  (7)  is  commonly  applied  to  steam  engines,  but  (8),  sometimes 
called  the  thermal  efficiency  at  brake,  or  economic  efficiency,  is  a  more 
correct  measure  of  economical  performance  for  any  heat  engine.  For 
engines  of  the  same  type,  when  the  mechanical  efficiency  is  -about  the 
same  for  the  usual  or  rated  load,  (7)  gives  a  fair  measure  for  comparison, 
but  for  widely  varying  types,  or  where  the  ratio  of  maximum  to  mean 
pressure  differs  greatly,  the  only  true  measure  of  economy  must  include 
the  friction  of  the  engine. 

There  is  no  indicated  horsepower  for  the  steam  turbine,  although  by 
assuming  mechanical  efficiency  the  so-called  turbine  horsepower  is  some- 
times calculated. 

For  any  type  of  prime  mover  driving  electrical  machinery  the  term 
electrical  horsepower  (e.h.p.  =  1.34  X  kilowatts)  is  used.  This  allows 
for  the  combined  friction  of  engine  and  generator.  Economy  may  then 
be  expressed  in  B.t.u.  per  e.h.p.-hr. 

The  most  common  method  of  expressing  economy  of  steam  engines  is 
in  Ib.  of  water  per  i.h.p.-,  or  per  kw.-hr.  This  may  be  used  for  the  com- 
parison of  simple  noncondensing  engines  using  saturated  steam,  but  is 
generally  unsatisfactory  and  is  being  replaced. by  the  heat-unit  method. 
To  show  the  discrepancy  of  basing  economy  upon  water  rate  alone,  as- 
sume engine  A  working  with  140  Ib.  absolute  pressure  and  exhausting  to 
atmosphere  with  a  steam  consumption  of  20  Ib.  per  i.h.p.-hr.;  while 
engine  B,  with  a  steam  pressure  of  165  Ib.  absolute  and  2  Ib.  absolute 
back  pressure  uses  12  Ib.  Assuming  initially  dry  saturated  steam  in 
both  cases,  the  B.t.u.  per  h.p.-hr.  for  A  is: 

20  X(1188  -  180)  =  20,160 


ECONOMY  73 

and  for  B: 

12  X(1193  -  94)  =  13,188. 

The  ratio  of  steam  consumption  of  A  to  that  of  B  is : 


and  of  heat  consumption: 


20,160  _ 
13,188 


which  is  nearly  14  per  cent.  less. 

41.  The  Heat  Factor. — It  is  obvious  from  the  T$  diagram,  that  if 
the  thermal  efficiency  of  the  Carnot  cycle  were  to  equal  unity  the  tem- 
perature of  exhaust  would  be  zero  absolute.  There  are  natural  limits 
which  make  this  physically  impossible,  and  even  with  the  best  of  our 
heat  engines  cause  the  efficiency  to  appear  pitiably  small.  As  the  greater 
part  of  this  seeming  deficiency  is  not  attributable  to  faulty  design  or 
construction,  it  is  fair  to  take  as  a  standard  of  comparison  a  cycle  which 
is  as  near  perfection  as  these  natural  limits  will  allow.  This  standard  is 
usually  the  Rankine  cycle  with  complete  expansion.  The  discarding 
of  adiabatic  compression  is  perhaps  not  necessitated  by  natural  physical 
conditions,  but  there  is  sufficient  practical  difficulty  attending  its  at- 
tempted application  to  warrant  its  rejection. 

The  Rankine  cycle  with  incomplete  expansion  is  sometimes  taken  as  a 
standard  when  the  terminal  pressure  of  the  actual  engine  is  known. 
This  seems  like  a  reasonable  practice  when  it  is  remembered  that  both 
capacity  and  economy  demand  a  certain  amount  of  terminal  drop; 
moreover,  it  is  the  true  measure  of  cylinder  efficiency. 

Denoting  the  efficiency  of  the  Rankine  cycle  by  eR,  the  ratio  of  effi- 
ciencies, variously  known  as  the  Rankine  cycle  ratio,  efficiency  ratio, 
and  heat  factor  (the  last  term  being  used  in  this  book),  is: 

F  =   °-  =  **  (9) 

eR        w 

In  applying  (9)  it  is  customary  to  take  e  as  the  indicated  thermal 
efficiency,  and  in  transferring  an  indicator  diagram  to  the  T$  diagram 
(see  Berry's  Temperature-entropy  Diagrams)  it  is  necessarily  so  assumed, 
but  there  is  no  reason  in  practical  work  why  e  should  not  be  based  upon 
net  work  performed,  or  b.h.p.,  for  steam  engines  as  well  as  for  internal- 
combustion  engines.  Practical  values  of  F  range  from  0.5  to  0.75. 

Cycle  design  consists  in  selecting  values  of  initial  pressure,  superheat, 
vacuum  and  ratio  of  expansion  which  will  give  the  highest  cycle 
efficiency  while  remaining  within  commercially  practical  limits.  In 


74  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

designing  the  engine,  proper  values  of  piston  speed,  rotative  speed  and 
cylinder  dimensions  must  be  selected  and  all  details  so  carefully  pro- 
portioned that  both  the  indicated  and  brake  efficiencies  of  the  engine 
shall  approach  as  nearly  as  possible  the  efficiency  of  the  theoretical 
cycle.  A  high  cycle  efficiency  combined  with  a  high  heat  factor  insures 
an  economical  engine,  as  the  product  of  the  two  equals  the  thermal 
efficiency,  which  after  all,  though  uncomplimentary,  is  the  absolute 
measure  of  economy  when  based  upon  the  brake  horsepower.  This 
statement  will  no  doubt  be  challenged,  as  -it  disregards  interest  on  invest- 
ment, depreciation  and  other  fixed  charges.  Exception  must  of  course  be 
made  for  temporary  installations,  and  when  the  fuel  is  all  refuse,  as  from 
wood-working  machinery,  there  being  often  a  surplus  to  dispose  of.  In 
this  case  the  engine  should  be  of  the  simplest  type.  However,  in  general, 
true  economy  must  always  tend  toward  the  conservation  of  those  natural 
resources  which  are  in  greatest  danger  of  depletion;  from  this  viewpoint, 
thermal  efficiency  will  be  the  criterion  of  economy. 

In  the  efficiency  of  the  Rankine  cycle  the  feed-water  temperature  is 
assumed  as  that  due  to  exhaust  pressure.  The  cycle  for  a  noncondensing 
plant  in  which  the  feed-water  temperature  is  lower  than  that  of  the 
exhaust  steam,  should  only  be  applied  as  the  ideal  cycle  when  the  boiler 
plant  is  included,  as  the  temperature  of  the  feed  water  has  no  influence 
on  the  economy  of  the  engine.  Then,  neglecting  the  steam  jacket  or 
reheating  receiver,  the  heat  factor  is: 

For  complete  expansion: 

e  2545  f 

F  =  *  =  5(C^C5 
For  incomplete  expansion  : 

2545 


,   (Pc  - 


The  heat  factor  for  complete  expansion  is  especially  useful  in  steam- 
turbine  design,  the  steam  consumption  being  reduced  to  turbine  horse- 
power. 

From  what  has  preceded  it  is  clear  that  in  reporting  economy  or  effi- 
ciency it  must  be  plainly  stated  whether  or  not  the  auxiliaries  are  included, 
and  whether  it  is  based  upon  i.h.p.  or  b.h.p.  In  comparing  engines  oper- 
ating under  different  conditions,  as  condensing  with  noncondensing,  or 
those  using  superheat  with  those  using  saturated  steam,  the  heat  con- 
sumption of  auxiliaries  required  for  any  given  mode  of  operation  should 
properly  be  included.  This  may  be  found  in  the  same  manner  as  for  the 
main  engine. 


ECONOMY  75 

Should  a  portion  of  the  exhaust  or  receiver  steam  be  utilized  for  heating 
or  industrial  purposes,  its  total  heat  content  may  be  deducted  from  that 
charged  to  engine  and  auxiliaries  on  the  ground  that  this  steam  must 
otherwise  have  been  furnished  by  the  boiler.  Should  all  of  the  exhaust 
steam  be  so  utilized  the  thermal  efficiency  would  then  equal  the  heat 
factor. 

On  the  other  hand  it  may  be  argued  that  the  manufacturing  plant 
is  utilizing  waste  heat  from  the  engine.  At  any  rate,  such  an  arrange- 
ment fosters  economy  and  is  often  a  deciding  factor  in  favor  of  the 
steam  engine. 

Should  all  of  the  exhaust  steam  be  required,  a  simple  type  of  engine 
with  small  ratio  of  expansion  could  be  employed.  This  would  deliver  a 
greater  weight  of  dryer  steam  and  possess  numerous  practical  advantages 
if  large  overload  capacity  were  not  required.  The  heat  factor  based  upon 
the  Rankine  cycle  with  incomplete  expansion  would  probably  be  greater, 
and  this  is  the  measure  of  economy  for  an  engine  working  under  these 
conditions.  Cylinder  condensation  and  radiation  losses  should  be  reduced 
to  a  minimum,  and  the  maximum  amount  of  heat  should  pass  out  with 
the  exhaust.  Such  an  engine  would  be  wasteful  only  upon  condition  that 
the  exhaust  steam  were  wasted.  In  some  cases  even  the  use  of  super- 
heated steam  would  be  well-advised,  should  this  provide  the  maximum 
heat  content  with  a  minimum  of  fuel  consumption. 

Theoretically,  the  heat  consumption  of  auxiliaries  is  an  insignificant 
fraction  of  that  required  by  the  main  engines,  but  may  practically  exert 
considerable  influence  upon  plant  economy.  The  use  of  uneconomical 
auxiliaries  is  never  justified  except  in  refuse-burning  plants,  even  though 
all  of  their  exhaust  steam  is  required  for  heating  feed  water. 

With  the  expection  of  that  which  refers  specifically  to  the  steam  engine, 
the  preceding  paragraphs  of  this  chapter  apply  as  well  to  the  steam 
turbine,  remembering  that  expansion  should  always  be  complete  in  the 
turbine. 

42.  Internal  Combustion  Engines. — A  common  method  of  expressing 
the  economy  of  gas-  and  oil  engines  is  in  cu.  ft.  of.  gas  or  Ib.  of  oil  per  i.h.p.- 
or  per  b.h.p.-hr.  For  producer-gas  engines  Ib.  of  coal  per  h.p.-hr.  is 
usually  given  in  stating  engine  performance.  This  practice  is  useful 
only  in  comparing  engines  using  the  same  fuel,  but  if  the  heating  value  of 
the  fuel  is  known  the  B.t.u.  per  h.p.-hr.  may  be  found.  If  the  fuel  is 
gas  the  heating  value  of  the  gas  must  be  based  upon  the  temperature  of 
the  gas  as  it  passes  through  the  meter,  or  both  volume  and  heating  value 
be  reduced  to  some  standard  temperature. 

In  the  A.S.M.E.  Code  the  higher  heating  value  of  the  fuel  is  used  as  a 


76  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

basis  for  computing  the  heat  charged  to  the  engine,  while  the  Vereines 
Deutscher  Ingenieure  employs  the  lower  heating  value.  In  view  of  this 
difference  of  opinion,  even  among  American  engineers,  all  test  reports 
should  state  whether  the  high  or  low  value  is  used.  The  use  of  the  lower 
value  may  be  justified  upon  similar  ground  that  the  heat  factor  of  the 
steam  engine  is  based  upon  the  Rankine  rather  than  the  Carnot  cycle, 
in  that  the  lowering  of  the  exhaust  temperature  to  a  point  where  the  latent 
heat  of  the  water  vapor  is  available  is  a  physical  impossibility.  On  the 
other  hand,  as  opinions  are  at  variance  concerning  the  proper  deduction 
to  be  made  in  determining  the  lower  heating  value,  greater  uniformity 
may  be  obtained  by  the  use  of  the  higher  value  until  there  is  a  more  general 
agreement  in  the  determination  of  the  lower  value,  which,  after  all,  must 
be  more  or  less  arbitrary. 

The  heating  value  being  determined,  the  heat  consumption  will  be: 

Lb.  fuel  per  hr.  X  heating  value  per  Ib.     .     . 
B.t.u.  per  h.p.-hr.  =  -  (12) 

horsepower 

or: 

Cu.  ft.  fuel  per  hr.  X  heating  value  per  cu.  ft.     ,     . 
B.t.u.  per  h.p.-hr.  =  -  — r-  (13) 

horsepower 

Then,  as  with  steam  engines  and  turbines,  the  thermal  efficiency  is: 

2545 (     . 

=  B.t.u.  per  h.p.-hr. 

Either  indicated-  or  brake  horsepower  may  be  taken  in  (12)  to  (14),  b.h.p. 
being  the  more  correct,  as  explained  already.  Both  values  are  sometimes 
given  in  test  reports.  In  this  connection  it  must  be  remembered  that 
indicator  diagrams  are  not  obtainable  from  engines  with  very  high  speed, 
and  that  a  brake  test  of  very  large  engines  is  not  always  feasible. 

43.  Efficiency  Ratio  of  Internal-combustion  Engines. — Actual  effi- 
ciency, either  indicated-  or  brake-  may  be  compared  with  the  theoretical 
cycle  efficiency.     This  is  given  by  (11)  and  (17),  Chap.  VI,  for  constant- 
volume  and  constant-pressure  cycles  respectively.     In  these  formulas  it 
is  usual  to  assume  air  as  the  working  substance,  with  the  corresponding 
value  of  the  ratio  of  specific  heats.     The  ratio  for  gas  mixture  theoreti- 
cally required  for  complete  combustion  is  slightly  different.     The  ratio 
of  actual  to  theoretical  efficiency  may  be  called  the  heat  factor  as  with 
steam  cycles.     It  is  not  used  for  design  as  with  steam  turbines  nor  is  it 
commonly  given  in  reporting  tests  as  with  steam  engines  and  turbines. 
This  may  in  part  be  due  to  lack  of  uniformity  in  assuming  conditions  for 
the  theoretical  cycle,  a  difficulty  not  experienced  with  the  ideal  steam 
cycle. 

44.  Comparative  Economy. — Any  extensive  collection  of  test  data,  or 


ECONOMY  77 

an  elaborate  discussion  of  the  relative  economy  of  the  different  heat 
engines  is  not  within  the  scope  of  this  book.  While  a  great  many  tests 
have  been  recorded  which  include  about  every  form  of  heat  engine  over  a 
wide  range  of  power,  the  results  for  any  given  class  vary  considerably 
among  themselves,  and  the  reduction  of  all  conditions  to  an  absolute 
standard  by  which  the  different  classes  may  be  compared  with  perfect 
fairness  is  nearly  impossible.  Best  efficiencies  differ  to  a  considerable 
extent  from  the  average  in  all  types,  and  tables  may  easily  be  compiled 
exhibiting  the  excellence  of  a  favorite  type  to  the  disparagement  of  its 
rival. 

It  may  be  said  for  the  steam  engine  and  the  steam  turbine,  that  under 
practically  the  same  working  conditions  for  machines  of  the  same  grade 
and  power,  their  maximum  thermal  efficiencies  are  the  same,  and  the 
selection  of  one  or  the  other  must  be  governed  by  other  conditions  for 
which  one  may  be  better  adapted  than  the  other. 

The  internal-combustion  engine  undoubtedly  has  a  higher  maximum 
thermal  efficiency  than  steam  machines.  There  is  less  difference  in  the 
efficiencies  of  large  and  small  powers  than  in  steam  engines  and  turbines, 
giving  the  advantage  to  the  internal-combustion  engine  for  small  and 
intermediate  powers. 

Some  data  of  heat-engine  performance  are  given  in  Chaps.  Ill  to  V. 
Economy  under  changing  loads  is  also  considered  in  Chaps.  XII  to  XV. 

Fuel  economy  is  not  the  only  factor  in  the  selection  of  an  engine.  In- 
terest on  investment,  rental,  depreciation  and  other  items  having  a  purely 
financial  bearing  are  sometimes  opposed  to  the  highest  fuel  economy  and 
govern  the  selection,  not  only  of  the  type,  but  the  grade  of  engine  within 
that  type.  This  condition  will  probably  continue  until  there  is  a  more 
acute  public  conscience,  or  a  new  era  of  economic  conditions  forces  upon 
us  a  more  careful  consideration  of  the  fuel  question. 


CHAPTER  IX 
CYLINDER  EFFICIENCY 

45.  Cylinder  efficiency  implies  losses  due  in  some  manner  to  the  cylin- 
der, and  suggests  practice  as  opposed  to  theory.     But  all  practice  has  its 
theory  even  though  it  be  unrecognized;  or,  due  to  its  complexity  we  are 
unable  to  follow  it  through  its  various  ramifications.     All  theory  is  not 
so  simple,  or  so  easily  checked  by  experiment  as  that  dealing  with  the 
effect  of  heat  upon  gases  and  vapors;  and  while  the  passage  of  heat 
through  metal  and  other  substances  has  been  theorized  and  experimented 
upon,  the  effect  of  the  metal  walls  of  a  cylinder  upon  a  vapor  or  gas  which 
is  going  through  a  certain  cycle  within  it  involving  changes  of  pressure 
and  temperature;  and  with  the-  duration  of  this  cycle  varying  from  two 
seconds  to  less  than  one-tenth  of  a  second;  is  indeed  difficult  to  analyze. 

Considering  the  conductivity  of  the  working  substance  and  of  the 
metal  of  the  cylinder,  and  the  fact  that  the  cylinder  is  jacketed  with  some 
heat-resisting  material,  steam  or  water;  also  the  impossibility  of  in- 
stantaneous temperature  measurements,  and  the  general  un-get-at- 
able-ness  of  the  whole  thing  for  reliable  observations;  about  the  best 
we  can  do  is  to  cover  the  entire  effect  with  the  heat  factor,  diagram  factor, 
or  the  thermal  efficiency  found  from  tests. 

It  seems  worth  while,  however,  by  the  application  of  simple  laws  to 
various  phases  of  engine  construction  and  operation,  to  study  the  ques- 
tion of  cylinder  losses  with  a  view  to  reducing  them. 

46.  The  Steam  Engine  Cylinder. — A  comparison  of  the  actual  with  the 
theoretical  steam  consumption  of  a  steam  engine  shows  that  much  more 
steam  enters  the  cylinder  than  is  required  to  do  the  work,  therefore  some 
of  it  must  be  condensed  in  its  passage  through  the  cylinder. 

Initial  Condensation. — Scientific  investigations,  notably  those  of  Him, 
of  the  behavior  of  saturated  steam  in  an  engine  cylinder,  have  established 
the  fact  that  nearly  all  of  the  condensation  occurs  as  the  steam  enters 
the  cylinder,  the  heat  being  withdrawn  by  the  walls  which  are  at  a 
temperature  much  below  that  of  the  steam.  This  is  known  as  initial 
condensation,  and  continues  up  to  the  point  of  cut-off. 

Condensation  and  Re-evaporation. — This  term  really  includes  initial 
condensation,  but  it  will  be  discussed  here  in  its  relation  to  the  remainder 

78 


CYLINDER  EFFICIENCY  70 

of  the  cycle,  beginning  at  cut-off.  Condensation  continues,  partly  due 
to  the  cooling  effect  of  the  cylinder  walls,  and  partly  to  the  conversion 
of  heat  into  work  which  would  occur  if  expansion  were  adiabatic.  This 
double  condensation  causes  the  expansion  curve  to  fall  below  the  adia- 
batic for  a  ways  as  may  be  seen  in  Fig.  54. 

The  transfer  of  heat  since  the  beginning  of  the  stroke  has  raised  the 
temperature  of  the  cylinder  walls,  so  that  the  temperature  of  the  steam 
as  it  expands  eventually  reaches  a  point  below  that  of  the  walls.     As  heat 
must   always  flow  from  a  higher  to  a 
lower    temperature,   the    steam   now 
begins  to  receive  heat  from  the  cyl- 
inder   walls.     This    begins    to   check 
further    condensation    and    in    some 
cases     will     re-evaporate    condensed 
steam    to   such   an   extent    that  the 
steam  is  dryer  at  the  end  of  expansion 
than  at  cut-off.     This  has  the  effect 
of   raising   the  expansion  line  above  FIG.  54. 

the  adiabatic  as  shown  in  Fig.  54. 

As  the  exhaust  valve  opens,  the  drop  in  pressure  due  to  free  expansion 
causes  further  drying.  The  cylinder  walls  continue  to  give  up  heat  to 
the  steam  during  the  exhaust  stroke  until  the  closure  of  the  exhaust 
valve. 

Compression  and  Clearance. — The  cylinder  walls  at  this  end  of  the 
cylinder  are  now  at  their  minimum  temperature,  and  the  steam  enclosed 
in  the  clearance  space  is  usually  assumed  to  be  dry.  Some  experiments, 
however,  indicate  that  clearance  steam  is  not  free  from  moisture  at  the 
beginning  of  compression;  this  is  probably  true  of  small  engines,  with 
which  most  of  the  compression  experiments  were  made.  Completing 
the  stroke  compresses  the  steam,  raising  its  pressure  and  temperature. 

Rankine's  cycle  with  clearance  shows  that  if  compression  raises  the 
pressure  in  a  nonconducting  cylinder,  so  that  it  equals  the  initial  pressure, 
and  if  expansion  is  complete,  clearance  does  not  effect  the  steam  consump- 
tion; but  expansion  is  not  complete  in  practice,  and  clearance  influences 
the  ratio  of  expansion  considerably,  the  effect  varying  with  changing 
cut-off.  The  effect  of  clearance  and  compression  is  also  dependent 
upon  other  factors. 

The  whole  effect  of  condensation  and  re-evaporation  so  far  has  been  to 
cool  the  cylinder  walls,  so  that  the  incoming  steam  for  the  next  cycle  will 
give  up  part  of  its  heat  by  condensation.  Re-evaporation  during  the 
latter  part  of  the  stroke  increases  the  work  done  slightly,  but  is  equivalent 


80  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

to  heat  added  at  a  temperature  less  than  the  maximum  and  is  therefore 
inefficient.  Assuming  no  compression,  and  that  the  exhaust  valve  closes 
at  the  end  of  the  stroke  enclosing  dry  steam  in  the  clearance  space,  initial 
condensation  will  be  due  to  the  comparatively  cool  walls  at  the  next 
admission  of  steam. 

It  is  probable  that  during  engine  operation,  the  inside  surface  of  the 
cylinder  walls  never  reaches  a  temperature  as  high  as  that  of  the  incoming 
steam,  or  as  low  as  that  of  the  exhaust  steam,  so  that  with  a  small  amount 
of  compression  the  temperature  of  the  steam  in  the  clearance  space  will 
not  be  as  high  as  the  cylinder- wall  temperature;  there  will  then  be  no 

further  loss  of  heat,  and  the  clearance 
steam  may  be  slightly  superheated. 
This  condition,  with  the  saving  of  the 
clearance  steam,  is  conducive  to 
economy. 

55  On  the  other  hand,  a  higher  com- 

pression may  raise  the  temperature  of 
the  steam  above  that  of  the  walls,  and  the  surface  of  the  walls  being  large 
in  proportion  to  the  weight  of  steam  in  contact  with  them  will  cause  the 
steam  to  condense,  lowering  the  pressure  as  shown  in  Fig.  55. 

It  is  well  known  that  the  presence  of  water  will  cause  steam  to  condense 
much  more  rapidly  than  contact  with  metal  of  the  same  temperature; 
therefore  if  compression  causes  condensation,  initial  condensation  for 
the  next  cycle  will  be  increased. 

There  has  been  much  discussion  about  compression  and  some  note- 
worthy experiments,  the  range  of  which,  however,  has  not  been 
comprehensive  enough  to  formulate  any  rules  governing  the  selection 
of  compression  pressures.  Mr.  Robert  R.  Fisher  in  Power,  July  29th, 
1913,  gives  the  results  of  tests  on  a  10  by  30  in.  simple,  noncondensing 
Corliss  engine,  which  showed  that  up  to  about  45  per  cent,  of  the  absolute 
initial  pressure,  compression  was  beneficial  to  economy.  The  effect  of 
speed  will  be  considered  presently. 

It  has  been  shown  by  certain  recent  experiments  that  the  percentage  of 
clearance  volume  has  little  effect  on  economy.  The  large  clearance, 
usually  considered  so  wasteful,  holds  a  larger  weight  of  steam  propor- 
tionate to  the  cooling  surface  of  the  walls  in  a  well-designed  cylinder, 
tending  to  reduce  initial  condensation.  An  earlier  compression  is  also 
necessary  to  get  a  given  cushion  effect;  this  encloses  a  greater  weight  of 
steam  which  better  resists  condensation  as  the  compression  temperature 
rises  above  that  of  the  walls.  The  influence  of  clearance  in  reducing 
relative  cooling  surface  is  more  noticeable  at  short  cut-off,  as  may  be  seen 


CYLINDER  EFFICIENCY  81 

in  Formula  1,  Par.  47.  The  effect  of  clearance  and  compression  and  their 
relation  to  cut-off  in  connection  with  theoretical  steam  consumption  is 
discussed  in  Chap.  XII,  Par.  62. 

The  cycle  just  completed  in  a  rather  roundabout  way  was  for  one  end  of 
a  cylinder  and  applies  to  a  single-acting  engine.  Most  steam  engines  are 
double-acting,  the  inlet  and  expansion  stroke  for  one  end  being  simul- 
taneous with  the  exhaust  stroke  for  the  other  end,  thus  complicating 
things.  From  a  consideration  of  the  foregoing,  it  is  obvious  that  the 
quantitative  prediction  of  results  by  any  mathematical  theory  however 
complex  and  apparently  complete,  is  futile  beyond  any  peradventure. 

Hirns  analysis,  and  later  the  temperature-entropy  diagram  have  been 
used  to  study  the  effect  of  cylinder  walls  on  saturated  steam,  and  these 
may  be  found  in  works  on  thermodynamics  and  the  temperature- 
entropy  diagram. 

47.  Factors  Affecting  Cylinder  Condensation. — These  include  practi- 
cally everything  that  must  be  considered  in  the  design  of  an  engine  of  a 
given  power,  and  with  few  exceptions  enter  into  the  design  of  all  steam 
engines.  These  exceptions  involve  special  construction  or  some  special 
application  of  heat  with  the  express  purpose  of  improving  economy. 


Factors  affecting  cyl- 
inder   condensation 


Common  to 
all  engines 


Speed. 

Size  of  cylinder. 

Design  of  cylinder. 

Range  of  pressure  and  temperature. 

Ratio  oi"  expansion. 


[  Steam  jacket. 
Special  <  Compounding. 

{  Superheated  steam. 

Speed. — Tests  of  a  given  engine  run  at  different  speeds  have  shown  that 
better  economy  is  obtained  at  the  higher  speeds.  At  high  speed  there  is 
less  time  for  heat  interchange  between  the  cylinder  walls  and  the  steam, 
and  the  minimum  temperature  is  probably  never  so  low. 

While  speed  is  a  factor  which  must  usually  be  considered  in  the  design 
of  the  engine,  the  economy  of  existing  engines  may  sometimes  be  im- 
proved by  increasing  the  speed  if  the  design  of  the  engine  will  permit  of 
the  resulting  increase  of  stress,  and  the  necessary  reduction  of  the  m.e.p. 
to  maintain  the  same  power  does  not  necessitate  too  short  a  cut-off,  the 
effect  of  which  will  be  explained  presently.  If  the  engine  is  belted  to  a 
jack  shaft,  the  pulley  on  this  shaft  must  be  increased  in  diameter  in  the 
same  ratio  that  the  engine  speed  is  increased,  to  retain  the  same  speed  of 
jack  shaft.  Should  the  increased  speed  cause  a  greater  velocity  of  the 


82  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

fly-wheel  rim  than  is  permissible — about  1  mile  per  minute — a  smaller 
fly-wheel  must  be  used,  retaining  the  original  jack-shaft  pulley.  The 
diameter  of  the  new  wheel  must  be  such  that  the  product  of  the  diameter 
and  r.p.m.  will  be  the  same  as  before  the  speed  change. 

Speed  change  is  sometimes  limited  by  the  area  of  ports  or  steam  pas- 
sages, which  if  too  small  may  cause  excessive  wire-drawing  and  offset  any 
gain  due  to  increase  of  speed. 

It  is  maintained  by  some  engine  builders  that  piston  speed  has  a  greater 
influence  on  economy  than  rotative  speed,  and  it  may  be  that  the  recent 
practice  of  higher  piston  speed  is  not  all  attributable  to  the  demand  for 
increased  capacity.  This  will  be  further  mentioned  under  cylinder 
design. 

Size  of  Cylinder. — For  cylinders  of  similar  form  the  volume  varies 
directly  as  the  cube  of  the  diameter,  while  the  surface  varies  as  the  square 
of  the  diameter;  this  may  be  seen  in  Table  2.  Therefore,  there  is  less 
cooling  surface  per  unit  volume  of  steam  in  large  cylinders  than  in  small, 
and  consequently  less  relative  condensation  in  a  given  time.  This  in 
part  accounts  for  the  superior  economy  usually  realized  in  large  engines 
even  though  the  rotative  speed  is  low.  The  piston  speed,  however,  is 
usually  fairly  high,  which  goes  to  support  the  suggestion  in  the  preceding 
paragraph;  there  have  also  been  exceptional  records  made  by  large  engines 
when  the  piston  speed  was  unusually  low. 

For  a  given  power  requirement  there  is  apt  to  be  a  conflict  between  high 
piston  speed  and  the  large  unit  idea,  especially  when  the  nature  of  the  ser- 
vice fixes  the  number  of  units.  A  compromise  will  usually  be  effected, 
governed  mostly  by  financial  considerations. 

Design  of  Cylinder. — This  includes  the  clearance  volume,  ratio  of 
stroke  to  diameter  of  cylinder,  and  the  jacket  or  covering.  With  the 
exception  of  special  designs,  such  as  the  uniflow  engine,  it  is  probably  safe 
to  state  that  the  clearance  volume  should  be  kept  as  small  as  possible. 
This  depends  upon  the  clearance  distance  between  piston  and  cylinder 
heads,  the  type  of  valves  used  and  the  general  arrangement  of  steam 
chest  and  ports,  which  must  all  be  carefully  worked  out  on  the  drawing 
board. 

The  ratio  of  stroke  to  cylinder  diameter  is  probably  more  often  deter- 
mined from  general  considerations  of  design  and  construction  than  from 
the  standpoint  of  economy.  A  large  ratio,  or  relatively  long  stroke  gives 
a  smaller  relative  amount  of  cooling  surface  for  the  same  cut-off  and  cylin- 
der diameter,  as  may  be  seen  from  Table  2,  and  permits  of  a  greater 
piston  speed  when  the  rotative  speed  is  limited  as  in  large  pumping 
engines,  or  is  fixed  by  direct  connection  to  electrical  machinery.  It 


CYLINDER  EFFICIENCY  83 

also  reduces  the  percentage  of  clearance  volume  for  the  same  piston  speed 
and  diameter  of  cylinder,  the  design  of  the  ports  being  the  same  for  a 
given  piston  speed. 

For  engines  of  the  same  power  and  rotative  speed  a  long  stroke  makes 
possible  a  greater  piston  speed  as  just  stated,  but  a  smaller  cylinder,  the 
latter  requiring  engine  parts  of  smaller  cross-sectional  area;  and  although 
the  long  stroke  necessitates  a  longer  engine,  the  smaller  diameter  of 
pins  and  bearings,  and  the  lighter  parts,  will  reduce  the  friction  and  in- 
crease the  mechanical  efficiency. 

The  relation  between  power,  cylinder  diameter,  rotative  and  piston 
speed,  may  be  readily  seen  in  the  equations  of  Chap.  XII,  from  which  a  bet- 
ter understanding  may  be  obtained  of  the  preceding  discussion.  From 
these  equations  it  is  apparent  that  for  an  engine  of  a  given  power,  assum- 
ing the.m.e.p.  to  be  the  same  in  any  case,  a  high  piston  speed  either  means 
a  high  rotative  speed  or  a  large  ratio  of  stroke  to  cylinder  diameter;  or 
that  both  rotative  speed  and  ratio  may  be  greater  than  with  a  lower 
piston  speed.  Although  the  relative  importance  of  these  two  factors 
may  not  be  definitely  stated,  they  are  both  conducive  to  economy  and  the 
advantage  of  higher  piston  speed  may  be  thus  explained. 

As  the  ratio  of  surface  to  volume  of  steam  undoubtedly  greatly  in- 
fluences condensation,  a  clearer  understanding  of  how  this  ratio  is 
affected  by  cylinder  diameter,  length  of  stroke  and  cut-off  will  be  of 
advantage,  so  a  simple  formula  will  be  derived.  Let: 

R  =  the  ratio  of  inside  cylinder  surface  in  one  end  of  a  cylinder  up  to 
cut-off,  to  the  volume  enclosed. 

D  =  the  diameter  of  the  cylinder  in  in. 

q  =  the  ratio  of  length  of  stroke  to  diameter  of  cylinder. 

I  =  ratio  of  the  portion  of  stroke  up  to  cut-off  to  the  entire  stroke. 

k  =  ratio  of  clearance  at  one  end  to  volume  "of  stroke,  neglecting 
counterbore  and  ports. 


Sarface  =  2      -  +  vD(l  +  k)qD 

TrD2 

Volume  =  —-  (I  +  k)qD 

=  surface       2(2q(l  +  k)  +  1] 
"  volume  q(l  +  k)D 

Assuming  a  clearance  of  4  per  cent.  Table  2  has  been  computed  from 
(1),  giving  values  of  R  for  values  of  D,  /  and  q  over  range  enough  to  show 
the  effect  of  each. 

The  influence  of  piston  speed  upon  the  design  of  engines  with  a  limited 
or  fixed  rotative  speed  is  not  apparent,  as  the  greater  piston  speed  neces- 


84 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


TABLE  2 


Values  of  R 

D 

in. 

9=1 

Q  =  2 

l  =  K 

i  =  y* 

i-H 

l  =  K 

z  =  K 

J  =  K 

10 

1.610 

1.090 

0.772 

1.010 

0.746 

0.586 

20 

0.805 

0.544 

0.386 

0.505 

0.373 

0.293 

30 

0.536 

0.363 

0.257 

0.336 

0.248 

0.195 

40 

0.402 

0.272 

0.193 

0.252 

0.186 

0.146 

sitates  a  smaller  cylinder  and  a  greater  ratio  of  stroke  to  diameter,  which 
have  opposite  effects  upon  the  value  of  R.  To  better  compare  these 
effects  examples  will  be  given  of  two  engines  of  different  power  assumed 
to  have  the  same  m.e.p.  and  cut-off,  but  each  having  several  different 
piston  speeds. 

The  rotative  speed  will  be  fixed  in  each  case  and  suited  to  the  power 
of  the  engine.  Cylinder  diameters  were  calculated  from  Formula  t(25), 
Chap.  XII,  from  which  the  following  notation  was  taken. 

P  =  m.e.p. 

N=  r.p.m. 

S  =  piston  speed  in  ft.  per  min. 
H  =  horsepower. 

L  =  length  of  stroke  in  in. 

I  and  q  are  used  as  in  Formula  (1),  from  which  R  is  calculated. 


Engine 

No.  1. 

P 

=  70, 

1  +  k  =  Y±,  H  =  200  and  N  = 

200. 

S  = 

600, 

L  = 

18", 

D 

=  14J4", 

q  = 

1 

.265, 

R 

= 

0.725 

S  = 

800, 

L  = 

24", 

D 

=  12^4", 

q  = 

1 

.960, 

R 

= 

0.660 

S  = 

1000, 

L  = 

30", 

D 

=  11", 

q  =  2.730, 

R 

as' 

0.630 

S  = 

1200, 

L  = 

36", 

D 

=  10", 

0  = 

3.600, 

R 

= 

0.622 

Engine 

No.  2. 

P 

=  70, 

1  - 

f  k  =  34 

H  -- 

=  1000  and 

N 

=  100. 

S  = 

600, 

L  = 

36", 

D 

-  31%", 

q  = 

1 

.135, 

R 

= 

0.348 

S  = 

800, 

L  = 

48", 

D 

=  27J^",     q  =  1.745, 

R 

= 

0.313 

.S  = 

1000, 

L  = 

60", 

D 

-  24}^", 

q  =  2.450, 

R 

S 

0.297 

S  - 

1200, 

L  = 

72", 

D 

=  22^",     q  =  3.200, 

R 

SB 

0.289 

It  is  obvious  that  the  gain  due  to  long  stroke  more  than  offsets  the 
loss  due  to  smaller  cylinder  as  the  piston  speed  becomes  greater,  giving  a 
decrease  in  the  ratio  R.  The  gain  per  hundred  feet  of  piston  speed  de- 
creases with  the  increase  of  piston  speed,  suggesting  a  practical  limit  for  a 
given  rotative  speed,  especially  if  it  leads  to  objectionable  construction, 


CYLINDER  EFFICIENCY  85 

A  comparison  of  the  values  of  R  for  engines  1  and  2  does  not  mean  that 
the  relative  condensation  per  Ib.  of  steam  used  is  indicated  thereby, 
because  this  is  counteracted  in  part  by  the  higher  rotative  speed  of  No.  1. 
R  is  but  one  of  the  factors  influencing  condensation,  and  it  may  be 
that  its  effect  is  proportional  to  some  power  of  this  factor  less  than 
unity. 

The  cross-sectional  area  of  the  engine  parts  is  less  for  the  smaller 
cylinder  diameter  as  previously  stated,  assuming  the  maximum  un- 
balanced pressure  to  be  the  same  in  all  cases;  and  while  this  is  offset  by 
increased  lengths  of  such  parts  as  are  affected  by  the  stroke,  making  the 
weight  of  these  parts  practically  the  same  in  either  case,  such  parts  as 
the  piston,  crosshead,  connecting  rod  ends,  crank  pin  and  shaft,  are 
only  affected  by  the  cylinder  diameter,  resulting  in  a  lighter  engine  as 
stated. 

Most  engine  cylinders  are  of  cast  iron,  which  is  a  good  conductor  of 
heat,  therefore  to  reduce  heat  loss  by  conduction,  radiation  and  convection, 
the  cylinder  must  be  covered  with  some  heat-resisting  material,  called  a 
nonconductor.  W°°cl  was  formerly  used,  but  as  higher  pressures  and 
temperatures  were  used  the  wood  was  replaced  by  asbestos  or  magnesia 
in  the  form  of  plaster,  which  is  more  easily  applied  and  more  satis- 
factory. This  is  covered  with  a  jacket,  or  lagging,  usually  of  sheet 
steel,  which  fulfils  the  purpose  of  a  finish  and  also  forms  a  dead  air 
space  which  aids  in  retaining  the  heat.  Steam  jackets  will  be  considered 
presently. 

Range  of  Pressure  and  Temperature. — If  the  difference  between  initial 
and  back  pressure  is  great,  there  will  be  a  correspondingly  great  tempera- 
ture difference  between  the  cylinder  walls  and  the  steam  at  the  moment 
of  admission.  As  this  is  responsible  for  initial  condensation,  it  follows 
that  the  greater  this  range  of  pressure  the  greater  will  be  the  condensa- 
tion. This  will  cause  loss  of  thermal  efficiency  unless  offset  by  some 
opposing  factor.  For  example,  increasing  the  steam  pressure  in  a  certain 
cylinder  increases  the  condensation,  but  with  a  proper  cut-off  the  gain 
in  work  done  due  to  the  increased  pressure  may  be  such  that  the  work 
per  given  weight  of  steam  is  more  than  with  the  lower  pressure.  Like- 
wise the  increase  of  pressure  range  by  lowered  back  pressure  due  to  con- 
necting a  condenser  may  produce  a  similar  result;  in  this  case  the 
economy  is  practically  always  better. 

Ratio  of  Expansion. — The  ratio  of  expansion  depends  upon  the  clear- 
ance and  the  cut-off.  For  an  engine  already  built  the  clearance  is  fixed, 
and  any  change  in  the  ratio  of  expansion  is  determined  by  the  cut-off. 
For  the  Rankine  cycle  of  maximum  economy  expansion  is  complete,  the 


86 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIG.  56. 


terminal  pressure  equalling  the  back  pressure.  Lowered  efficiency  re- 
sults if  the  ratio  is  decreased  or  increased,  the  latter  resulting  in  a  loop 
in  the  diagram,  giving  negative  work  as  in  Fig.  56. 

Complete  expansion  would  also  give  the  highest  efficiency  in  actual 
engines  if  it  were  not  for  cylinder  condensation  and  friction.  Neglecting 
the  latter  for  the  present,  it  is  obvious  from  Table  2  that  if  the  rate  of 

condensation  of  a  given  weight  of 
steam  depends  upon  the  surface  to 
which  it  is  exposed — other  things 
being  equal — the  relative  condensation 
is  greater  at  short  cut-off.  Thus,  a 
cut-off  must  be  chosen  practically 
always  greater  than  will  give  complete 
expansion,  and  such  that  the  sum  of  the  loss  due  to  incomplete  expansion 
and  that  due  to  condensation  is  a  minimum.  This  cannot  be  definitely 
predetermined,  and  for  a  given  engine  may  only  be  found  by  test ;  but 
it  is  usually  from  J£  to  }£,  depending  upon  the  various  factors  already 
discussed  and  to  follow. 

Obviously  this  most  economical  cut-off  must  be  such  as  to  give  the 
highest  thermal  efficiency  at  brake,  thus  providing  for  friction.  It  is  also 
desirable  that  it  correspond  to  the  load  to  be  carried  by  the  engine  most 
of  the  time,  or  to  the  rated  horsepower,  and  this  is  usually  aimed  at  in 
steam-engine  design.  Then  by  change  of  cut-off  the  engine  can  adapt 
itself  to  changes  of  load  from  zero  brake  horsepower  to  50  per  cent,  over- 
load, and  in  some  cases  as  high  as  100  per  cent.,  the  economy  decreasing 
as  the  load  varies  either  side  of  the  rated  load,  assuming  that  this  was 
correctly  chosen.  A  typical  steam  consumption  diagram  is  shown  in 
Fig.  57.  The  effect  of  initial  ^ -i 
condensation  shows  in  a  marked 
degree  at  the  smaller  powers. 

Steam  Jacket. — A  form  of 
cylinder  construction  in  which 
steam  surrounds  the  cylinder 
barrel  and  heads  is  sometimes 
used  to  improve  economy. 
The  steam  is  usually  supplied 
from  the  steam  main  which 
supplies  the  engine,  the  condensed  steam  being  piped  to  a  trap.  The 
object  of  this  steam  jacket  is  to  keep  the  cylinder  walls  at  a  higher  tem- 
perature throughout  the  cycle  and  thus  reduce  initial  condensation.  The 
jacket  steam  replaces  part  of  the  heat  carried  out  with  the  exhaust  due  to 


01   *" 

1_ 

I15 

3I03 

X 

^v^>» 

•""-  _ 

—  —  — 

,      - 

„«—  —  •"' 

0     40      50      60      TO      80      90      100     IIC 
I.H.P. 
FIG.  57. 

CYLINDER  EFFICIENCY 


87 


re-evaporation,  and  supplies  that  carried  away  by  radiation,  which  would 
otherwise  be  taken  from  the  working  steam. 

While  the  consumption  of  steam  in  the  cylinder  is  reduced  by  the 
application  of  the  steam  jacket,  the  jacket  consumption  partly  offsets 
this  result,  so  that  the  increase  in  economy  is  much  less  than  would  other- 
wise be  possible.  In  some  instances  there  is  no  gain  whatever  and  some- 
times an  actual  loss,  so  that  the  steam  supply  to  the  jackets  is  discontinued. 

It  is  obvious  that  engines  in  which  the  design  or  working  conditions 
tend  to  increase  condensation  are  more  greatly  benefited  by  steam  jackets, 
the  gain  in  some  instances  being  as  high  as  30  per  cent.  Thus,  engines  at 
light  load  with  short  cut-off  show  considerable  gain,  with  less,  or  no  gain 
at  a  more  economical  cut-off,  and  sometimes  a  loss  at  overloads. 

Comparatively  few  engines  are  equipped  with  steam  jackets,  their 
most  usual  application  being  to  large  pumping  engines  and  marine  engines 
of  large  power.  Their  diminishing  use  is  probably  partly  due  to  the 
increasing  use  of  superheated  steam. 

Compounding. — The  general  principle  of  compound  engine  operation 
is  explained  in  Par.  3,  Chap.  III.     After  expansion  is  effected  in  the  high- 
pressure  cylinder,  the  steam  is  exhausted  to  a  receiver,  from  whence  it  is 
admitted  to  a  low-pressure  cyl- 
inder  and  again  expanded,   ex- 
hausting finally   to   the  atmos- 
phere    or     a     condenser.      In 
triple-    or    quadruple-expansion 
engines    this   operation   is  con- 
tinued through  one  or  two  more 
cylinders.     The     advantage    of 
compounding  may  be  explained 
with    two-stage    expansion    and 
the  following  discussion  will  be 
so  confined. 

Assuming  a  compound  engine  to  operate  on  the  Rankine  cycle  with 
complete  expansion,  and  that  for  the  sake  of  simplicity  the  receiver  is 
indefinitely  large,  so  that  the  transfer  of  steam  to  and  from  the  receiver 
does  not  change  the  pressure,  the  pressure-volume  diagram  is  shown  in 
Fig.  58.  Pursuant  with  current  practice  let  the  receiver  pressure  divide 
the  diagram  into  two  equal  parts,  equally  dividing  the  work  between 
the  two  cylinders.  This  may  be  done  by  dividing  the  difference  between 
the  heat  content  at  pi  and  p3  at  the  same  entropy,  by  two,  subtracting 
it  from  the  heat  content  at  pi  and  finding  from  an  entropy  chart  or  table 
the  pressure  corresponding  to  the  resulting  heat  content. 


FIG.  58. 


88 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


For  example,  assume  adiabatic  expansion  between  151  Ib.  per  sq.  in. 
absolute  to  14.7  Ib.  absolute,  at  1.52  entropy.  From  table  or  chart,  Table 
3  may  be  constructed. 

TABLE  3 


Pressure 

Temp.,  °  F. 

Heat  content 

Specific  volunne 

Pl  =  151.0 
p2  =    50.8 
p,  =    14.7 

359 

282 
212 

1153.3 
1072.2 
991.1 

vi  =    2.859 
v,  =    7.465 
w3  =  22.39 

Thus  for  a  simple  engine  operating  between  the  same  pressures,  the 
temperature  range  from  admission  to  exhaust  is  147  degrees  F.,  while  for 
the  compound  it  is  77  and  70  degrees  in  the  high-  and  low-pressure 
cylinders  respectively;  or  approximately  one-half  of  what  it  would  be  in 
the  single  cylinder,  thus  reducing  the  initial  condensation. 

Referring  to  Fig.  58,  the  ratio  of  expansion  for  a  simple  engine  would 
be  vi/vi,  or  7.84.  For  the  compound  engine  it  is  v*/vi,  or  2.61  in  the  high- 
pressure  cylinder,  and  vz/v2,  or  3  in  the  low-pressure.  For  the  simple 
diagram  assumed,  the  cut-offs  are  the  reciprocals  of  these  numbers. 
Thus,  while  the  total  ratio  of  expansion  is  the  same  in  both  simple  and 
compound  engine,  and  theoretically  the  most  economical,  in  the  actual 
engine  the  longer  cut-offs  in  the  cylinders  of  the  compound  would  result 
in  less  initial  condensation  than  the  shorter  cut-off  in  the  simple  engine. 

For  the  Rankine  cycle,  assuming  as  it  does  a' nonconducting  cylinder, 
there  would  be  no  advantage  in  compounding  from  a  thermodynamic 
standpoint;  but  it  serves  to  illustrate  the  effect  of  compounding  upon 
temperature  range  and  ratio  of  expansion,  two  very  important  factors 
influencing  condensation. 

Clearance,  compression,  the  receiver,  and  the  fact  that  the  expansion 
is  not  adiabatic,  alter  the  values  of  Table  3  to  some  extent  when  applied 
to  actual  engines;  these  will  be  considered  in  Chap.  XIII. 

Cylinder  Ratio  and  Terminal  Drop. — These  quantities  are  interdepend- 
ent when  a  nearly  equal  division  of  work  is  desired,  and  the  question  of 
their  proper  values  to  insure  the  best  economy  has  been  considerably 
discussed.  A  knowledge  of  the  principles  of  Chap.  XIII  is  necessary  for 
an  intelligent  consideration  of  this  subject,  and  a  previous  study  of  the 
same  is  assumed. 

Then  it  is  apparent  that  within  reasonable  limits,  the  power  of  a 
compound  engine  working  with  a  given  ratio  of  expansion  is  mainly 
dependent  upon  the  size  of  the  low-pressure  cylinder,  the  high-pressure 
cylinder,  or  the  ratio  of  the  volume  of  stroke  of  the  low-pressure  to  that 


CYLINDER  EFFICIENCY 


89 


of  the  high-pressure  cylinder,  having  little  influence.  That  this  ratio 
does  affect  the  terminal  pressure  in  the  high-pressure  cylinder  is  plainly 
seen  in  Fig.  59,  in  which  V2  is  the  volume  of  stroke  of  the  low-pressure 
and  Vi  of  the  high-pressure  cylinder.  Volume  V\  is  for  a  low  cylinder 
ratio,  or  value  of  Vz/Vi,  while  TV,  shown  by  the  dotted  lines,  is  for  a  high 
cylinder  ratio.  Terminal  drop  p  is  for  the  low  ratio  and  p'  the  high 
ratio. 

As  elsewhere  explained,  a  certain  amount  of  terminal  drop  is  necessary 
for  practical,  economic  engine  operation,  and  this  is  provided  for  in  the 
low-pressure  cylinder  by  the 
assumption  of  a  total  ratio  of 
expansion  which  is  expected  to 
give  the  maximum  economy  at 
the  rated  load. 

Although  engines  with  high 
cylinder  ratios  have  been  built 
for  a  good  many  years,  the  lower 
ratios  are  in  the  majority,  a 
very  common  value  of  the  ratio 
being  four.  Though  there  have 
been  high-economy  records  with 
both  low  and  high  ratios,  advo- 
cates of  the  high  ratio  claim 
superior  economy  when  working  conditions  are  the  same  in  both 
cases. 

In  Fig.  59  it  will  be  noticed  that  the  total  area  is  some  less  for  the 
high  ratio.  On  the  other  hand  the  cut-off,  especially  in  the  high-pressure 
cylinder,  is  longer.  With  a  high  terminal  drop  the  steam  will  be  dryer 
as  it  enters  the  receiver  than  with  low  drop,  if  the  quality  is  the  same  at 
terminal  pressure. 

For  the  purpose  of  comparison,  let  three  engines  be  considered,  each  to 
have  a  rated  indicated  horsepower  of  500,  neglecting  the  diagram  factor; 
a  piston  speed  of  1000  ft.  per  min.,  and  the  same  stroke  and  r.p.m. 
Also  let  the  initial  and  back  pressures  be  140  and  2  Ib.  absolute,  clearance 
4  per  cent.,  compression  0.8  stroke  in  the  low-pressure  cylinder,  and  in  the 
high-pressure  cylinder  such  that  the  two  compression  curves  lie  on  the 
same  curve.  Assume  the  curves  of  expansion  and  compression  to  be 
hyperbolas  and  that  the  total  ratio  of  expansion  is  30  in  all  cases. 

The  cylinder  ratios  are  4,  5.55  and  7.56;  cylinder  diameters  were 
computed  by  the  formulas  of  Chap.  XIII  as  was  also  the  theoretical 
water  rate.  The  temperature  range  was  taken  from  steam  tables  for  the 


FIG.  59. 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


pressures  used  in  each  case.  The  quality  is  given  to  show  the  condition 
the  steam  may  be  in  at  terminal  pressure  in  the  high-pressure  cylinder 
to  become  dry  due  to  free  expansion  by  the  time  it  reaches  the  receiver 
pressure.  These  values  are  given  in  Table  4.  The  cylinder  ratios 
for  engines  2  and  3  given  in  the  table  are  not  those  used  in  the  calculation, 
but  are  the  result  of  rounding  up  the  calculated  diameter  to  even  inches. 
The  ratio  R  was  found  by  Formula  (1). 

TABLE  4 


Cyl.  diam. 

Cut-off 

Temp,   range 

Ratio  R 

Relative  effect 

Cyl. 

Qual- 

Water 

ratio 

ity 

rate 

H.p., 
in. 

L.p., 
in. 

H.p. 

L.p. 

H.p. 

L.p. 

H.p. 

L.p. 

H.p. 

L.p. 

4.00 

16 

32 

0.0985 

0.231 

135 

105 

0.653 

0.331 

0.999 

7.76 

1.000 

1.000 

5.55 

14.0 

33 

0.14170.233 

136 

104 

0.591 

0.329 

0.994 

7.78 

1  .  024 

1.025 

7.56 

12.0 

33 

0.1980 

0.247 

138 

102 

0.567 

0.316 

0.986 

7.97 

1.063 

1.067 

By  a  study  of  Table  4  and  the  preceding  paragraphs  of  this  chapter 
it  is  evident  that  the  economy  is  affected  inversely  as  the  ratio  R,  the 
temperature  range,  the  quality  and  the  theoretical  water  rate.  Assuming 
the  product  of  a  constant  and  the  reciprocals  of  these  quantities  as  unity 
for  each  cylinder  of  engine  1,  their  relative  combined  influence  may  be 
found  and  is  given  in  the  last  two  columns  of  Table  4.  This  indicates 
an  average  gain  in  economy  of  the  highest  over  the  lowest  ratio  of  6.5 
per  cent. 

The  influence  of  the  receiver  and  piping  upon  temperature  range  and 
condensation  have  been  neglected,  it  being  assumed  the  same  in  each  case. 

Lack  of  comprehensive  experimental  data,  and  the  complex  nature  of 
the  problem  make  impossible  the  construction  of  a  formula  including 
these  tabulated  quantities  which  may  be  used  to  predict  economy,  but 
it  seems  reasonable  to  believe  that  they  give  the  direction  in  which  im- 
proved economy  may  be  looked  for. 

Superheated  Steam. — 'When  superheated  steam  enters  the  cylinder  of 
a  steam  engine,  heat  is  transferred  to  the  walls  as  with  saturated  steam, 
but  instead  of  immediate  condensation  upon  contact,  which  still  further 
augments  condensation,  the  superheat  must  first  be  withdrawn.  Mean- 
while, the  cylinder  walls  have  increased  in  temperature,  so  that,  should 
the  steam  reach  the  saturation  point  before  cut-off  occurs,  the  tendency  to 
condensation  is  greatly  reduced.  If  superheating  is  carried  far  enough 
condensation  will  not  begin  until  after  cut-off,  and  even  be  delayed  until 
expansion  is  partly  completed.  Engineers  differ  as  to  the  degree  of 
superheat  which  may  be  used  to  advantage,  some  maintaining  that 


CYLINDER  EFFICIENCY  91 

economy  increases  with  the  amount  of  superheat,  while  a  number  con- 
servatively place  the  limit  at  about  100  degrees  F.,  having  consideration  of 
the  damage  to  castings  and  the  difficulty  of  lubrication  attending  the  use 
of  exceedingly  high  temperatures.  Even  a  small  amount  of  superheat  is 
of  great  advantage,  for  though  some  condensation  begins  at  the  walls 
before  the  point  of  cut-off  is  reached,  superheated  steam  is  a  much 
poorer  conductor  of  heat  than  saturated  steam,  so  that  the  bulk  of  the 
steam  is  but  little  influenced. 

The  effect  of  superheat  on  economy  is  largely  indirect,  the  actual  gain 
often  being  twice  that  given  by  the  Rankine  cycle. 

In  addition  to  the  reduction  of  initial  condensation,  increased  specific 
volume  in  part  accounts  for  steam  economy,  a  cubic  foot  of  steam  at 
140  Ib.  absolute  pressure  and  100  degrees  superheat  weighing  but  86  per 
cent,  of  saturated  steam  at  the  same  pressure.  This  is  in  part  offset  by 
the  fact  that  the  expansion  curve  falls  below  tha-t  of  saturated  steam, 
necessitating  a  larger  volume  at  cut-off  to  do  the  same  work. 

Superheat  probably  affects  several  of  the  factors  previously  discussed, 
as  it  offsets  the  influence  of  initial  condensation;  however,  the  features 
of  design  best  adapted  to  the  economical  use  of  saturated  steam  will  prob- 
ably produce  the  best  results  with  superheated  steam.  It  seems  probable, 
however,  that  due  to  the  reduction  of  condensation  at  short  cut-off,  the 
most  economical  cut-off  will  be  shorter,  making  it  possible  to  carry  a 
desired  overload  without  the  usual  long  cut-off  and  loss  of  expansive 
work.  On  the  other  hand,  as  just  stated,  the  expansion  curve  for  super- 
heated steam  is  said  to  drop  more  rapidly  than  that  of  saturated  steam, 
necessitating  a  longer  cut-off  to  obtain  the  same  terminal  pressure.  This 
effect  is  perhaps  more  noticeable  with  high  degrees  of  superheat;  in  fact, 
the  author  has  failed  to  notice  much  difference  in  this  respect  in  any  super- 
heated-steam  diagrams  that  have  come  to  his  notice,  these  having  been 
mostly  for  a  moderate  degree  of  superheat. 

The  range  of  pressure  may  be  increased  with  superheat,  making  the 
condensing  simple  engine  more  desirable  than  with  saturated  steam. 
The  tendency  has  been,  however,  where  high  pressure  has  been  employed 
with  saturated  steam,  as  in  locomotives,  to  reduce  the  pressure  when 
superheat  is  employed. 

While  there  are  engines  specially  built  for  superheated  steam,  it  is 
advantageously  applied  to  existing  engines  with  various  types  of  valves. 
This  is  especially  true  of  locomotives,  the  addition  of  the  superheater 
increasing  capacity  as  well  as  economy. 

In  the  Journal  of  the  A.S.M.E.,  Jan.,  1916,  Mr.  Robt.  Cramer  points 
out  the  theoretical  advantage  of  using  steam  of  high  pressure  with  mod- 


92  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

erate  superheat,  over  lower  pressure  with  greater  superheat.  He  makes 
his  comparison  between  pressures  of  200  and  600  Ib.  per  square  inch, 
limiting  the  temperature  in  both  cases  to  600  degrees  F.  This  allows 
superheat  of  218  and  113  degrees  respectively.  A  common  pressure  for 
compound  engines  and  steam  turbines  is  150  Ib.  gage,  and  a  common 
range  of  superheat  is  from  100  to  150  degrees.  Using  the  nearest  values 
without  interpolation  from  Peabody's  entropy  table,  comparison  with  a 
gage  pressure  of  250  Ib.  is  given  in  Table  5,  assuming  a  vacuum  of  28  in. 
of  mercury  and  nearly  equal  maximum  temperature.  Efficiency  is  that 
of  the  Rankine  cycle,  given  by  Formula  (8),  Chap.  VII. 

TABLE  5 


Absolute  pressure      

164  800 

264  300 

Maximum,  temperature              .      .              .... 

508  000 

503  900 

Superheat 

142  000 

97  900 

Entropy                                       

1  650 

1  590 

Efficiency 

0  292 

0  328 

The  higher  pressure  shows  a  gain  of  12  per  cent.,  and  while  this  may 
not  be  practically  realized,  there  probably  would  be  a  substantial  increase 
in  efficiency,  as  the  superheat  is  sufficient  to  greatly  reduce  if  not  prevent 
all  condensation.  The  gain  is  obviously  due  to  the  reception  of  a 
larger  percentage  of  heat  at  high  temperature.  There  is  no  practical 
difficulty  in  the  use  of  250  Ib.  steam  from  the  standpoint  of  either  boiler  or 
engine,  and  it  is  probable  that  much  higher  pressures  may  be  practical. 
This  question  has  been  much  discussed  since  the  appearance  of  Mr. 
Cramer's  paper. 

High  pressure  shows  greater  theoretical  gain  than  high  superheat, 
which  may  be  readily  understood  by  a  study  of  the  entropy  diagram; 
it  is  probable,  however,  that  where  extremely  high  pressures  are  used, 
high  superheat  will  be  a  desirable,  if  not  a  necessary  accompaniment, 
especially  in  steam  turbine  operation,  as  otherwise,  condensation  will 
begin  too  early. 

High  steam  pressures  are  no  novelty,  500  Ib.  having  been  used  on 
naval  vessels  and  as  high  as  1000  Ib.  in  automobiles.  A  steam  car  has 
recently  been  developed  to  carry  600  Ib.  pressure  with  no  superheat. 
The  advisability  of  carrying  such  pressures  in  stationary  steam  plants  is 
problematical  and  has  a  number  of  noted  authorities  as  its  advocates. 
The  question  is  probably  a  financial  one.  If  gain  in  thermal  efficiency 
and  capacity  offsets  interest  on  investment  and  insurance,  high  pressures 
will  be  used,  the  matter  of  safety  having  no  more  bearing  than  with  high- 
tension  electric  currents  or  munition  factories. 


CYLINDER  EFFICIENCY  93 

While  no  definite  rules  have  been  given  for  obtaining  maximum 
cylinder  efficiency,  a  careful  consideration  of  the  foregoing  principles, 
combined  with  a  reasonable  amount  of  practical  experience  and  common 
sense,  should  aid  the  designer  in  his  compromise  of  conditions  to  secure 
good  working  results. 

48.  The  Internal-combustion  Engine  Cylinder.— Gas-  and  oil-engine 
cycles  are  discussed  in  Chaps.  V  and  VI.  Due  to  the  high  maximum 
temperature  and  great  temperature  range,  the  theoretical  efficiency 
is  high  compared  to  that  of  the  steam  engine  and  turbine.  The  cumula- 
tive effect  of  the  high  temperature  upon  the  metal  walls  of  the  cylinder 
is  such  that  the  metal  is  heated  to  a  point  where  it  will  not  be  abl  to 
withstand  the  pressure  if  some  means  are  not  employed  to  withdraw  part 
of  the  heat  as  fast  as  it  is  generated.  This  is  most  commonly  accom- 
plished by  the  water  jacket,  but  sometimes  air  cooling  is  used  with  small 
engines.  Heat  passes  from  the  working  gases  through  the  cylinder  walls 
into  the  cooling  medium,  and  as  the  rate  of  heat  flow  depends  upon  tem- 
perature difference,  more  heat  is  withdrawn  at  combustion  and  during 
the  early  part  of  expansion  than  during  the  remainder  of  the  cycle,  and 
this  lowers  the  maximum  temperature. 

Unlike  the  steam  engine,  the  temperature  of  the  working  fluid  during 
exhaust  never  becomes  as  low  as  that  of  the  cylinder  walls.  The  next 
charge  then  receives  from  the  walls  a  portion  of  this  heat  of  combustion 
at  much  less  than  the  maximum  temperature.  This  addition  of  heat  to 
the  charge  causes  a  proportionately  higher  temperature  during  the  com- 
pression stroke,  so  that  for  a  given  compression  pressure,  the  high  tem- 
perature at  the  time  of  ignition  causes  a  higher  temperature  along  the 
combustion  line  with  its  correspondingly  greater  heat  transfer.  The 
heat  added  to  the  charge,  by  increasing  its  specific  volume,  reduces  its 
weight;  this  decreases  the  capacity  of  the  engine,  increasing  the  relative 
friction,  and  the  ratio  of  cooling  surface  to  weight  of  charge — also  promot- 
ing heat  transfer. 

It  is  commonly  stated  that  the  quantity  of  heat  withdrawn  from  the 
engine  cylinder  should  be  the  smallest  possible  amount  consistent  with 
practical  operation;  but  while  the  theoretical  efficiency  of  an  internal- 
combustion  engine  is  independent  of  the  initial  temperature  of  the  charge, 
it  is  not  probable  that  actual  brake  efficiency  would  be  improved  by  entire 
absence  of  cooling,  even  though  lubrication  and  the  strength  of  the  metal 
were  not  impaired. 

Cylinder  dimensions,  speed,  etc.,  affect  heat  transfer  to  cylinder  walls 
in  the  same  way  as  in  the  steam  engine,  but  the  problem  is  an  entirely 
different  one.  Prevention  of  heat  transfer  is  the  aim  in  the  steam  engine, 


94 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


while  in  the  internal-combustion  engine  a  compromise  between  thermal 
efficiency,  capacity  and  lubrication  must  be  effected.  It  is  obvious  that 
the  cooling  system  should  be  flexible,  so  that  it  may  be  adjusted  to  give 
the  best  operating  results.  In  certain  engines  the  temperature  of  the 
cooling  water  is  controlled  by  a  thermostat  which  may  be  set  at  the  proper 
temperature,  which  Js  found  by  experience. 

As  previously  stated,  cooling  the  cylinder  has  little  influence  upon 
economy  within  reasonable  limits.  If  the  heat  is  not  taken  out  in  the 
cooling  water,  combustion  is  suppressed  by  dissociation  as  extremely 
high  temperature  is  reached  and  the  heat  is  thrown  out  in  the  exhaust. 
As  engines  increase  in  size  the  ratio  of  cooling  surface  to  volume  of 
cylinder  decreases,  the  natural  result  being  a  larger  proportion  of  heat  in 
the  exhaust.  This  is  shown  by  the  first  three  engines  in  Table  6,  taken 
from  Peabody's  Thermodynamics.  The  last  engine  used  blast-furnace 
gas  and  was  liberally  cooled  with  water. 
The  amount  of  cooling  water  required  may  be  found  as  follows : 

TABLE  6 


Distribution  of  heat 

Engine 

size, 

in. 

Work 

Jacket 

Exhaust 

6.75  X 

13.70 

0.16 

0.52 

0.32 

9.50  X 

18.00 

0.22 

0.44 

0.35 

26.00  X 

36.00 

0.28 

0.24 

0.48 

51.20  X 

55.13 

0.28 

0.52                     0.20 

Let  h    =  heating  value  per  cubic  feet  of  gaseous  fuel  or  per  pound  of 

liquid  fuel  in  B.t.u. 

w    =  cubic  feet  of  gaseous  fuel  or  pound  of  liquid  fuel  per  b.h.p.-hr. 
W    =  pound  of  cooling  water  per  b.h.p.-hr. 
G    =  gallons  of  cooling  water  per  b.h.p.-hr. 
TR    =  Temperature  rise  in  water  in  degrees  F. 

q    =  fraction  of  total  heat  supply  withdrawn   by   cooling   water. 
eB    =  thermal. efficiency  at  brake. 
Equating  the  heat  withdrawn  with  the  heat  taken  by  the  water: 

qwh  =  WTR 
Also  from  (8),  Chap.  VIII: 

2545  7       2545 

eB  = 


wh 


eB 


Substituting  gives: 


2545g 
eB 


CYLINDER  EFFICIENCY  95 

or: 


W  =  and  0  = 


If  q  =  0.35,  eB  =  0.20   and    TR  =  70,  then:  W  =  61    and  G  =  7.5. 

As  a  practical  suggestion  it  has  been  stated  that  if  jacket  water  is  run 
through  the  jackets  for  some  time  after  the  engine  is  shut  down,  it  will 
prevent  the  heating  of  the  water  remaining  in  the  jacket  to  the  point 
where  salts  held  in  solution  will  be  precipitated,  thus  forming  deposits  on 
the  walls,  causing  overheating.  It  will  also  prevent  the  evaporation  of 
oil  from  the  piston  pin. 

Lack  of  agreement  among  authorities  concerning  combustion  in  the 
cylinders  of  internal-combustion  engines,  the  properties  of  gases  at  high 
temperatures  and  the  effect  of  this  upon  combustion  makes  the  discussion 
of  these  subjects  seem  out  of  place  in  a  book  of  this  character.  Much  of 
value  of  this  nature  may  be  found  in  Giildner's  Internal-combustion 
Engines,  to  which  the  reader  is  referred. 

Guldner  suggests  a  greater  ratio  of  stroke  to  diameter  than  has  been 
customary,  and  greater  piston  speeds,  both  of  which  are  being  adopted  in 
present-day  practice,  especially  in  automobile  engines.  Their  effect  upon 
capacity  and  weight  is  perhaps  greater  than  upon  economy  of  operation. 
High  piston  speed  involves  either  large  valves  or  high  gas  velocities. 
Guldner  recommends  a  gas  velocity  of  4500  ft.  per  min.  witn  a  maximum 
of  6000  ft.  under  the  most  favorable  conditions.  With  a  piston  speed  of 
1000  ft.  per  min.  and  a  valve  lift  of  J£  its  diameter  —  which  is  given  as  a 
maximum  by  Guldner  —  the  valve  diameter  would  be  about  0.55  and  0.47 
of  the  cylinder  diameter  for  the  two  values  of  gas  velocity  respectively. 
This  exceeds  usual  practice  at  the  present  time.  The  piston  speed  of  Gtild- 
ner's  engine  given  in  his  book  is  730  ft.  per  min.  At  maximum  valve 
lift  given  by  him  and  the  valve  diameter  scaled  from  his  reproduction 
of  a  working  drawing  of  the  engine,  the  gas  velocity  would  be  8000  ft. 
per  min.  A  discussion  of  this  subject  with  examples  from  present 
practice  is  given  in  Chap.  XX. 

As  in  the  steam  engine,  the  tendency  is  toward  higher  rotary  and 
piston  speeds.  A  piston  speed  of  2000  ft.  per  min.  is  not  uncommon  in 
automobile  engines  and  3200  ft.  has  been  used  in  racing  engines.  Rotary 
speeds  from  1500  to  3000  r.p.m.  are  also  used.  Although  gas  velocities 
are  undoubtedly  increased  by  these  high  speeds,  with  a  consequent  loss, 
there  is  still  a  gain  in  capacity. 

Valve  timing  plays  an  important  part  in  both  capacity  and  efficiency 
and  is  discussed  in  Chap.  XX.  The  influence  of  compression  is  con- 
sidered in  Par.  76,  Chap.  XIV. 


PART  III— FRICTION  AND  LUBRICATION 


CHAPTER  X 

MECHANICAL  EFFICIENCY 

Notation. 

HB      =  brake  horsepower  (b.h.p.)  in  general. 

HBR    =  b.h.p.  at  rated  load. 

HI      =  indicated  horsepower  (i.h.p.)  in  general. 

HIR    =  i.h.p.  at  rated  load. 

Hf      =  friction  horsepower  (f.h.p.). 

e         =  mechanical  efficiency  at  any  power. 

eR       =  the  same  at  rated  load. 

P        =  maximum  unbalanced  pressure  in  engine  cylinder  in  pounds 

per  square  inch. 

PM     =  mean  effective  pressure  (m.e.p.)  in  pounds  per  square  inch. 
fji         =  coefficient  of  friction  of  engine  if  PM  =  P- 

49.  A  general  expression  for  mechanical  efficiency  of  engines  is  given  in 
Formula  (7),  Chap.  VI.  The  mechanical  efficiency  changes  with  the  load- 
ing and  is  obviously  zero  when  all  external  load  is  removed  and  the  engine 
is  only  overcoming  its  own  friction.  The  horsepower  developed  under 
these  conditions  is  known  as  the  friction  horsepower,  and  the  m.e.p., 
the  friction  m.e.p. 

Friction  depends  upon  many  factors,  such  as  the  condition  of  the 
rubbing  surfaces,  the  system  of  lubrication  and  the  lubricant  used,  the 
relation  of  work  done  to  the  weight  of  the  moving  parts;  and,  other 
things  being  equal,  it  is  usually  relatively  less  for  a  given  type  of  engine 
as  the  size  is  increased. 

It  is  therefore  apparent  that  no  fixed  rule  can  be  given  by  means  of 
which  friction  may  be  determined  with  accuracy  in  all  cases,  but  due  to 
the  very  conditions  named,  it  is  convenient  for  the  purpose  of  arranging 
and  studying  the  results  of  tests,  to  derive  formulas  giving  relations 
between  indicated,  brake  and  friction  horsepowers  for  different  condi- 
tions of  loading. 

To  be  intelligible,  mechanical  efficiency  must  always  be  stated  with 
reference  to  some  given  load;  this  may  be  the  maximum  load  or  fraction 
7  97 


98  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

thereof,  or  any  rated  load.  The  efficiency  will  then  be  different  for 
any  other  load. 

The  friction  horsepower  usually  increases  with  increase  of  load;  the 
increase  is  often  slight,  however,  and  for  practical  purposes  the  f.h.p. 
may  be  assumed  constant.  This  assumption  will  be  made  in  the  present 
discussion  and  the  results  compared  with  actual  values  from  practice. 

In  what  follows,  H  will  denote  horsepower  and  e  mechanical  efficiency. 
The  subscripts  B  and  /  denote  brake  and  indicated  horsepower  respec- 
tively. The  subscript  R  refers  to  rated  power;  with  steam  engines  this 
is  usually  the  most  economical  power,  allowing  for  from  50  to  100  per 
cent,  overload,  depending  upon  the  type  of  valve  gear  used.  For  steam 
turbines  and  internal-combustion  engines,  though  at  or  near  the  load 
giving  best  economy,  the  rated  load  is  more  nearly  the  maximum;  in 
most  cases,  however,  from  10  to  20  per  cent,  overload  may  be  carried. 

When  subscript  R  is  not  used  in  conjunction  with  B  and  /,  any  other 
than  the  rated  power  is  meant.  The  f.h.p.  is  assumed  constant  for  a 
given  engine.  For  convenience  of  expression  let  : 

^  =  fc  (1)     and  --q  (2) 


The  expression  for  mechanical  efficiency  at  any  rated  load  may  be 
written  : 


HBR  HBR  1 


HIR  HBR  +  HF  ..      i       Hf 

From  (3) : 


(3) 


-  =  -     -  1  =  l   -^  (4) 

BR  CR  6R 


Efficiency  at  any  load  in  terms  of  b.h.p.  is: 

HB  1 


P  H,  (5) 


From  (1),  (4)  and  (5): 

1_  _1_ 

6  —  jj-  1  /£>\ 

i  J-    g*          i  4.  1  ~  e«  W 

h  kHBK  keR 

Dividing  (1)  by  (2)  gives: 

k         HIR      H  B          HB     HJR   _     6 
q         Hj      HBR         HI    HBR         VR 

p 

from  which  k  =  q  •  -  (7) 


MECHANICAL  EFFICIENCY 


99 


To  express  efficiency  in  terms  of  i.h.p.,  substituting  (7)  in  (6)  and 
solving  for  e: 


Should  e  be  known  at  some  other  than  the  rated  load,  eR  may  be 
found  from  (8);  or: 

eR  =  1  -  q(l  -  e}  (9) 

Formulas  (6)  and  (8)  give  the  efficiency  for  any  load  when  it  is  known 
at  the  rated  load,  when  the  fraction  of  the  load  is  given  in  terms  of  b.h.p. 
and  i.h.p.  respectively. 

Results  of  tests  are  given  in  the  following  tables.  Tables  7  to  9  are 
from  Goodman's  Mechanics  'Applied  to  Engineering,  and  are  from  tests 
on  an  experimental  steam  engine  with  different  methods  of  lubrication. 

The  data  in  Tables  10  to  12  are  for  Westinghouse  vertical  4-cycle, 
3-cylinder  gas  engines  using  natural  gas,  and  were  taken  from  Giildner's 
Internal-combustion  Engines. 

TABLE  7.  —  SYPHON  LUBRICATION 


Hj 

2.75 

9.250 

10.230 

11.140 

12.340    13.950 

14.290 

HB 

0.00 

5.630 

7.500     7.660 

9.090 

11.090 

11.250 

HF 

2.75 

3.630 

2.730 

3.480 

3.250 

2.860 

3.040 

e 

0.00 

0.608 

0.733 

0.688 

0.738 

0.795 

0.788 

TABLE  8. — SYPHON  AND  PAD  LUBRICATION 


Hz 

2.48 

5.160 

6.830 

8.300 

11.500 

13.840 

17.020 

22.300 

HB 

0.00 

2.350 

3.940 

5.610 

8.700 

10.820 

13.890 

19.090 

fo 

2.48 

2.810 

2.890 

2.690 

2.800 

3.020 

3.130 

3.210 

e 

0.00 

0.455 

0.578 

0.676 

0.756 

0.783 

0.806 

0.857 

TABLE  9. — FORCED  LUBRICATION 


ffl 

49.800 

102.700 

147.600 

193.600 

217.500 

HB 

44.500 

97.000 

140.600 

186.000 

209  .  500 

HF 

5.300 

5.700 

6.500 

7.600 

8.000 

e                     0.912 

0.945 

0.955    . 

0.962 

0.964 

TABLE  10. — 25-  BY  30-iN.  GAS  ENGINE 


e 

Nominal  load 

Hi 

HB 

HF 

N 

k 

Actual 

Gale. 

H 

262.5 

207.5 

55.0 

151.7 

0.375 

0.791 

0.754 

H 

449.3 

383.8 

65.5 

150.0 

0.693 

0.855 

0.850 

Rated 

621.4 

553.1 

68.3 

149.2 

1.000 

0.891 

0.891 

Max.              676.7 

605.6 

71.1 

146.7 

0.095 

0.895 

0.894 

100  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  11. — 19-  BY  22-iN.  GAS  ENGINE 


Nominal  load 

Hi 

HB 

HF 

Ar 

k 

e 

Actual 

Calc. 

K 

147.2 

121.3 

25.9 

206.3 

0.507 

0.825 

0.780 

Rated 

273.5 

239.0 

34.5 

203  .  0 

1.000 

0.875 

0.875 

IX 

365.5 

325.3 

40.2 

198.0 

1.360 

0.890 

0.862 

TABLE  12. — 13-  BY  14-iN.  GAS  ENGINE 


e 

Nominal  load 

Hi 

HB 

HF 

N 

k 

Actual 

Calc. 

0 

18.57 

0.00 

18.57 

265.5 

0.000 

0.000 

0.000 

H 

53.69 

30.81 

22.88 

264.0 

0.273 

0.574 

0.534 

H 

85.55 

64.37 

21.18 

264.0 

0.570 

0.752 

0.706 

H 

114.69 

90.25 

24.44 

260.0 

0.800 

0.787 

0.770 

Rated 

139.73 

112.86 

26.87 

258.0 

1.000 

0.808 

0.808 

Max. 

164.22 

143.17 

21.05 

256.8 

1.270 

0.873 

0.842 

TABLE  13. — HORNSBY — AKROYD  OIL  ENGINE 


e 

Nominal  load 

HI 

HB 

HF 

N 

m 

Actual 

Calc. 

H 

13.10 

9.  CO 

4.10 

203.0 

0.336 

0.687 

0.677 

H 

22.40 

17.96 

4.44 

202.4 

0.672 

0.802 

0.807 

Rated 

32.15 

27.74 

4.41 

202.6 

1.000 

0.862 

0.862 

TABLE  14. — GULDNER  GAS  ENGINE 


Nominal 
load 

m 

HB 

HF 

h 

e 

Actual 

Calc. 

H 

18.85 

11.15 

7.70 

0.317 

0.592 

0.550 

H 

22.20 

13.30 

8.90 

0.378 

0.600 

0.594 

y* 

26.40 

17  .  65. 

8.25 

0.502 

0.668 

0.660 

H 

31.10 

21.50 

9.60 

0.612 

0.692 

0.704 

H 

37.40 

25.90 

11.50 

0.737 

0.692 

0.741 

Rated 

44.20 

35.10 

9.10 

1.000 

0.795 

0.795 

Table  13  contains  data  given  by  Gtildner  for  a  Hornsby-Akroyd  oil 
engine. 

Table  14  is  for  a  Giildner  gas  engine. 

From  these  data  it  may  be  seen  that  for  practical  use  in  designing, 
formulas  (6)  and  (8)  may  be  used.  It  may  also  be  seen  that  the  value  of 


MECHANICAL  EFFICIENCY  101 

e  for  a  given  load  increases  with  the  size  of  the  engine,  both  for  steam 
and  gas;  but  it  is  considerably  greater  for  the  steam  engine  in  proportion 
to  the  power.  It  is  less  for  Diesel  engines  that  for  gas  engines;  W.  H. 
Adams,  Trans.  A.S.M.E.,  vol.  37,  p.  460,  gives  0.75  for  4-cycle  engines 
and  0.70  for  2-cycle  engines.  The  Hornsby-Akroyd  oil  engine,  Table  13, 
has  a  rated  load  efficiency  of  0.862,  although  developing  only  18  b.h.p., 
but  the  compression  was  only  about  45  to  50  lb.,  and  the  explosion  pres- 
sure only  180  to  200  lb. 

There  seems  to  be  some  relation  between  pressure  and  mechanical 
efficiency  for  engines  having  about  the  same  m.e.p.  When  the  ratio  of 
maximum  unbalanced  pressure  to  m.e.p.  is  high,  the  engine  parts  must 
be  heavier,  and  while  theoretically,  friction  work  depends  upon  mean 
rather  than  maximum  pressure,  high  maximum  pressures  seem  to  inter- 
fere with  lubrication;  at  any  rate,  the  friction  m.e.p.  increases  with  the 
ratio  of  maximum  to  mean  pressure. 

The  formula  f  OP  mechanical  efficiency  may  be  written  : 


An  empirical  expression,  considering  the  pressure  ratio  just  mentioned, 
may  be  written: 

Hp  P  HB  /i  1  \ 

=  M     +  m  - 


in  which  P  is  maximum  unbalanced  pressure  and  PM  the  m.e.p.  The 
coefficient  of  friction  /*  may  be  assumed  as  the  ratio  of  friction  to  useful 
work  if  the  pressure  were  uniform  throughout  the  stroke,  or  if  P  =  PM; 
its  value  may  be  taken  from  0.02  to  0.06.  HB  may  be  taken  as  the  b.h.p. 
of  one  cylinder  end  and  m  a  constant  depending  upon  the  type  of  engine. 
Assume  for  average  values,  the  following: 

P  P 

For  simple  steam  engines  ............................   p—  =  2  and  /z  p—  =  0  .  08. 

P  P 

For  compound  steam  engines  ........................    p—  =  2.5  and  /z  p—  =  0.  10. 

P  P 

For  gas  —  or  other  constant-vol.  engines  ...........  ....   p-  =  4  and  /x  p—  =  0.  16. 

P  P 

For  Diesel  engines  .................................  p-  =  6  and  ^  p—  =  0.  24. 

Values  of  m  for  steam  engines  and  4-cycle  internal-combustion  engines 
are  given  in  Table  15.  These  are  rather  arbitrary  and  from  0.05  to  0.15 
may  be  added  for  2-cycle  engines,  the  larger  values  being  for  small,  high- 
speed engines. 


102 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Formulas  (10)  and  (11)  may  be  used  with  judgment  for  obtaining  a 
fair  approximation  to  ordinary  values  found  in  practice,  but  of  course 
wide  deviations  will  be  found.  A  wider  study  of  data  would  probably 
result  in  more  accurate  constants,  or  perhaps  a  better  formula. 

TABLE  15. — VALUES  OF  m 


Type  of  engine 


Steam-small,  with  unbalanced  valve 

Steam-compound 

Steam-locomotive 

2-cylinder,  single-acting,  internal-combustion 

3-cylinder,  single-acting,  internal-combustion 

4-  and  8-cylinder,  single-acting,  internal-combustion 

6-  and  1 2-cylinder,  single-acting,  internal-combustion 

1-cylinder,  double-acting,  internal-combustion 

2-cylinder,  double-acting,  twin  internal-combustion ........ 

2-cylinder,  double-acting,  tandem  internal-combustion 

4-cylinder,  double-acting,  twin-tandem  internal-combustion 
Small  engines  with  1  or  2  cylinders,  poorly  attended 


0.07 
0.02 
0.08 
0.02 
0.03 
0.04 
0.06 
0.02 
0.04 
0.05 
0.06 
0.04 


Reference 
Mechanics  applied  to  engineering,  Prof.  Goodman. 


CHAPTER  XI 

LUBRICATION 

Notation. 

d     =  diameter  of  bearing  in  inches. 

I     =  length  of  bearing  in  inches. 

k     =  ratio  of  length  to  diameter  =  l/d. 

Ds  =  diameter  of  standard  simple  engine  cylinder,  designed  for  some 
standard  pressure.  (See  Par.  63,  Chap.  XII  and  Par.  72, 
Chap.  XIII.) 

a  =  area  of  rubbing  surface  of  bearing  in  square  inches;  this  is  the 
projected  area  (Id)  for  cylindrical  bearings. 

p    =  total  mean  load  on  bearing  in  pounds. 

p  —  mean  pressure  in  pounds  per  square  inch;  for  cylindrical  bear- 
ings, per  square  inch  of  projected  area. 

pM  =  maximum  pressure  in  pounds  per  square  inch. 

S     =  shearing  stress  in  turbine  shaft  in  pounds  per  square  inch. 

V    =  velocity  of  rubbing  surface  in  feet  per  minute. 

N    =  r.p.m. 

M  =  speed  in  knots,  or  nautical  miles  per  hour. 

H    =  horsepower  of  engine  (i.h.p.). 

h     =  heat  in  B.t.u.  per  hour  developed  in  bearing. 

C,  K  and  ra  are  constants  in  formulas. 

50.  General  Principles. — Friction,  lubrication  and  the  proportioning 
of  bearing  surfaces  are  so  intimately  related  in  connection  with  the 
design  of  heat  engines  that  they  may  be  considered  together  in  one  chapter. 
The  preceding  chapter  dealt  with  the  effect  of  friction  upon  power,  and 
later  chapters  will  give  the  general  construction  of  bearings,  while  in  the 
present  chapter  a  study  of  some  of  the  factors  influencing  friction,  and 
means  of  reducing  it  will  be  attempted. 

The  laws  of  friction  are  exceedingly  complicated,  and  while  much 
valuable  experimental  work  has  been  done,  and  most  lubricating  problems 
are  practically  handled,  no  comprehensive  mathematical  treatment 
has  yet  been  presented.  References  to  valuable  material  are  given  at 
the  end  of  this  chapter. 

Proper  lubrication  is  effected  when  a  thin  film  of  any  lubricant  lies 
between  two  surfaces  in  such  a  way  that  there  is  no  metallic  contact. 
There  is  then  no  appreciable  wear.  Without  lubrication,  there  is  wearing 

103 


104 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


of  the  surfaces,  and  with  great  unit  pressures  the  surfaces  may  usieze, " 
or  become  welded  together,  so  that  if  further  movement  is  forced  the 
surfaces  become  torn  and  "scored." 

There  may  be  all  stages  between  no  lubrication  and  perfect  lubrica- 
tion, with  behavior  varying  accordingly;  for  this  reason  it  is-  well  to 
consider  the  laws  of  dry  as  well  as  of  lubricated  surfaces.  The  laws  of 
friction  of  dry  surfaces  credited  to  Morin  have  been  revised  in  the  light 
of  more  recent  experiments  and  are  given  by  Goodman  in  Mechanics 
Applied  to  Engineering,  in  parallel  with  the  laws  of  lubricated  surfaces, 
and  they  will  be  given  here. 


DRY  SURFACES 

1.  The  f rictional  resistance  is 
nearly  proportional  to  the  normal 
pressure  between  the  two  surfaces. 


2.  The  f  rictional  resistance  is 
nearly  independent  of  the  speed  for 
low  pressures.  For  high  pressures  it 
tends  to  decrease  as  the  speed  in- 
creases. 


3.  The  frictional  resistance  is  not 
greatly  affected  by  temperature. 


4.  The  frictional  resistance  de- 
pends largely  upon  the  nature  of  the 
material  of  which  the  rubbing  sur- 
faces are  composed. 


LUBRICATED  SURFACES 

1.  The    frictional   resistance   is    almost 
independent    of    the    pressure    with    bath 
lubrication  so  long  as  the  oil  film  is  not 
ruptured,  and  approaches  the  behavior  of 
dry    surfaces    as    the   lubrication  becomes 
meager. 

2.  The  frictional  resistance  of  a  flooded 
bearing,  when  the  temperature  is  artificially 
controlled,   increases    (except  at  very  low 
speeds)    nearly    as   the   speed,    but  when 
the    temperature    is    not    controlled    the 
friction    does    not   appear   to   follow   any 
definite  law.     It  is  high  at  low  speeds  of  rub- 
bing,   decreases    as    the    speed    increases, 
reaches  a  minimum  at  a  speed  dependent 
upon  the  temperature  and  the  intensity  of 
pressure;    at   higher   speed    it    appears   to 
increase  as  the  square  root  of  the  speed ;  and 
finally,    at   speeds   of   over    3000  feet  per 
minute,  some  believe  that  it  remains  con- 
stant. 

3.  The     frictional    resistance    depends 
largely  upon  the  temperature  of  the  bearing, 
partly  due  to  the  viscosity  of  the  oil,  and 
partly  to  the  fact  that  the  diameter  of  the 
bearing  increases  with  a  rise  of  temperature 
more  rapidly  than  the  diameter  of  the  shaft, 
and  thereby  relieves  the  bearing  of  side 
pressure. 

4.  The     frictional     resistance     with     a 
flooded  bearing  depends  but  slightly  upon 
the  nature  of  the  material  of  which  the 
surfaces  are  composed,  but  as  the  lubrication 
becomes  meager,  the  friction  follows  much 
the  same  laws  as  in  the  case  of  dry  surfaces. 


LUBRICATION 


105 


DRY  SURFACES 

5.  The  friction  of  rest  is  slightly 
greater  than  the  friction  of  motion. 


6.  When  the  pressures  between 
the  surfaces  become  excessive,  seizing 
occurs. 


7.  The    frictional    resistance    is 
greatest  at  first,  and  rapidly  decreases 
with  the  time  after  the  two  surfaces 
are  brought  together,  probably  due  to 
the  polishing  of  the  surfaces. 

8.  The    frictional    resistance    is 
always  greater  immediately  after  re- 
versal of  direction  of  sliding. 


LUBRICATED  SURFACES 

5.  The  friction   of  rest  is   enormously 
greater  than  the  friction  of  motion,  espe- 
cially if  thin  lubricants  be  used,  probably  due 
to  their  being  squeezed  out  when  standing. 

6.  When    the    pressures    between    the 
surfaces  become  excessive,  which  is  at  a 
much  higher  pressure  than  with  dry  sur- 
faces,   the  lubricant   is  squeezed  out  and 
seizing  occurs.     The  pressure  at  which  this 
occurs  depends  upon  the  viscosity  of  the 
lubricant. 

7.  The  frictional  resistance  is  least  at 
first,   and  rapidly  increases  with  the  time 
after  the  two  surfaces  are  brought  together, 
probably  due  to  the  partial  squeezing  out  of 
the  lubricant. 

8.  Same  as  in  the  case  of  dry  surfaces. 


15      50     75      100     115    r50    175    200 
Rubbing  Speed -Ff. per  Sec. 

FIG.  60. 


IIS 


The  way  in  which  friction  varies  with  velocity  for  lubricated  surfaces 
is  shown  in  Fig.  60  taken  from  Goodman.  A  comparison  of  curves  A 
and  B  also  shows  the  influence  of  temperature  as  stated  in  Law  3  for 
lubricated  surfaces.  A  comparison  of  curves  C  and  Z),  and  of  curves  E 
and  F  also  shows  the  effect  of  pressure. 

It  is  usually  considered  that  friction  and  wear  are  so  closely  related 
as  to  be  nearly  synonymous  terms,  but  experiments  at  low  pressures  show 
that  for  dry  surfaces  the  friction  is  less  than  for  a  bearing  flooded  with 
oil,  but  it  is  certain  the  wear  must  be  greater.  The  friction  of  a  lubricated 


106  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

bearing  is  due  to  the  shearing  of  the  lubricant,  which  does  not  damage  the 
bearing  although  heat  is  produced  with  a  rise  in  temperature. 

From  Law  3  of  lubricated  surfaces,  a  hot  bearing,  within  reasonable 
limits,  is  more  efficient  than  a  cool  one,  as  the  thinning  of  the  lubricant 
due  to  temperature  reduces  the  friction;  this  is  also  shown  by  curves  A 
and  5,  Fig.  60. 

Martin  gives  an  instance  of  a  steam  turbine  bearing  in  which  the  heat 
generated  at  90  degrees  F.  was  double  that  generated  at  165  degrees,  show- 
ing less  friction  at  the  high  temperature.  He  also  states  that  certain  steam 
turbine  bearings  running  at  a  temperature  of  195  degrees  F.  have  given  no 
trouble.  High  temperature  is  normal  for  a  properly  lubricated  high- 
speed bearing,  but  the  oil  must  be  kept  sufficiently  viscous  to  keep  the 
rubbing  surfaces  apart.  The  greater  the  normal  viscosity  of  the  lubricant, 
the  higher  will  the  temperature  be,  due  to  greater  shearing  resistance,  and 
the  greater  the  allowable  bearing  pressure  at  a  given  speed. 

It  is  not  advisable  to  run  a  bearing  at  its  least  friction,  involving  as  it 
does  high  temperature  with  a  less  viscous  condition  of  the  lubricant,  as 
an  increase  in  temperature  from  any  cause  may  still  further  decrease  the 
viscosity  so  that  the  oil  film  will  not  be  dragged  completely  around  the 
bearing.  To  control  the  temperature,  some  bearings  are  constructed  so 
that  cold  water  may  be  circulated  around  them,  and  in  steam  turbine 
practice  with  forced  lubrication,  the  oil  is  passed  through  a  cooler  before 
being  fed  to  the  bearings. 

The  temperature  of  steam  turbine  bearings  is  not  all  due  to  the  shear- 
ing of  the  oil,  but  in  part  to  the  transmission  of  heat  of  the  steam  through 
the  shaft,  so  a  bearing  temperature  of  180  degrees  F.  is  probably  perfectly 
normal  for  a  turbine  if  there  is  no  foaming  of  the  oil. 

A.  G.  Christie,  Trans.  A.S.M.E.,  vol.  34,  p.  435, 

I     I      I    J  states   that  if  the  bearings  of  a  steam  turbine  are 

-*—  flooded  with  oil  at  100  degrees  F.  so  that  its  tempera- 
ture upon  leaving  is  about  125  degrees,  the  life  of 
the  oil  is  much  longer  than  when  very  hot  oil  is 
used,  and  the  wear  of  the  bearings  absolutely  pre- 
vented. He  further  says  that  the  use  of  water- 
FIG.  61.  cooled  bearings  is  to  be  discouraged,  water  being 

much  better  employed  in  an  oil  cooler. 

A  light  oil,  with  ample  continuous  feed,  properly  filtered  and  cooled 
will  probably  give  minimum  friction  with  maximum  life  of  oil  and 
bearings. 

With  oil-bath  lubrication  as  in  Fig.  61,  oil  is  drawn  up  around  the  bear- 
ing as  shown,  the  layer  being  thicker  on  the  "on"  side  than  on  the  "off" 


LUBRICATION 


107 


side.     This  has  been  shown  to  be  so  by  actual  experiment  by  Goodman, 
and  that  in  case  of  wear,  it  was  greater  on  the  off  side. 

Experiments  by  Tower  also  show  that  the  oil  pressure  varies,  being  a 
maximum  nearly  midway  between  the  on  and  off  sides,  and  greater  on  the 
off  side  than  on  the  on  side.  He  also  shows  that  to  feed  oil  at  the  center 
where  the  pressure  is  greatest  causes  the  oil  to  flow  out  rather  than  in. 
The  results  of  some  of  Tower's  tests,  given  by  Goodman  (Mechanics 
Applied  to  Engineering),  are  given  in  Table  16,  which  shows  the  seizing 
pressure  and  coefficient  of  friction  for  different  methods  of  supplying 
lubricants,  and  for  different  kinds  of  bearings.  Bath  lubrication  gives  the 
best  results  and  may  be  taken  as  the  standard  of  lubrication.  Equiva- 
lent methods  are  those  in  which  the  bearing  is  flooded  with  oil,  or  fias 
stream  feed,  such  as  some  of  the  forced  feed  systems. 

TABLE  16 


Form  of  bearing 

0 

0 

0 

0 

O 

i 

M, 

J^ 

Seizing        load, 
Ib.  per  sq.  in. 
Coefficient      of 

100 

150 

370 

550 

600 

200 

90 

friction  

0.01 

0.01 

0.006 

0.006 

0.001 

0 

.01 

0.035 

TABLE  17 


Mode  of  lubrication 

Relative  friction 

Tower 

Goodman 

Bath  

1.00 

1.00 
1.32 
2.21 
4.20 

Saturated  pad 

Ordinary  pad  

6.48 
7.06 

Syphon.  . 

Pad  lubrication,  giving  next  best  results,  is  accomplished  by  having 
the  journal  in  contact  with  oil-soaked  pads.  Waste  in  railway  journal 
boxes  is  used  for  this  purpose.  A  comparison  between  methods  of 
applying  lubricant  is  given  in  Table  17,  from  Goodman. 

51.  Lubricants  and  Their  Application. — As  acid  is  produced  by  the 
decomposition  of  animal  and  vegetable  oils,  they  are  little  used  for  heat 
engine  lubrication,  except  as  the  former  may  be  combined  in  small 


108  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

proportions  with  mineral  oil  to  form  cylinder  oil  for  saturated  steam, 
promoting  better  retention  of  the  oil  on  the  surfaces.  Mineral  oils  may 
then  be  considered  as  the  lubricants  for  heat  engines,  and  a  few  important 
characteristics  will  be  mentioned.  Further  information  may  be  found 
in  Gill's  Engine  Room  Chemistry  and  in  treatises  on  Testing  and  Ex- 
perimental Engineering. 

Viscosity,  or  "body, "  is  the  degree  of  fluidity  or  internal  friction  of 
an  oil.  This  varies  with  the  temperature  and  must  be  tested  at  some 
standard  temperature,  commonly  70  degrees  F.  for  engine  oils  and  212 
degrees  F.  for  cylinder  oils.  The  least  viscous  oil  which  will  stay  in  place 
and  do  its  work  should  be  used,  as  it  absorbs  less  power  in  friction. 
Viscosimeters  for  testing  the  viscosity  of  oils  are  commonly  constructed 
upon  the  principle  of  the  rate  of  flow  through  a  standard  orifice,  and  the 
results  are  given  in  seconds.  The  longer  it  takes  for  a  given  quantity  of 
oil  to  flow  through  the  orifice,  the  greater  the  viscosity.  When  viscosity 
is  given,  the  viscosimeter  should  be  named. 

Density  or  specific  gravity  of  mineral  oils  is  usually  measured  with  the 
Baume  hydrometer,  and  expressed  in  degrees  Baume  (degrees  B.).  The 
specific  gravity  is  found  by  the  formula : 

144.3 
Specific  gravity  ==  ^3  +  defr  B; 

The  cold  test  is  the  temperature  at  which  the  oil  will  just  flow.  In 
stationary  power  plants  this  is  of  little  importance,  but  for  exposed  work 
in  cold  climates  it  must  not  be  overlooked. 

The  flash  point  is  the  temperature  at  which  oil  gives  off  vapors  in 
sufficient  quantity  to  explode  when  mixed  with  air,  but  the  oil  will  not 
continue  to  burn. 

The  fire  test  or  burning  point  is  the  temperature  at  which  vapor 
enough  is  given  off  to  burn  continuously. 

The  gumming  test  is  to  ascertain  the  amount  of  change  taking  place  in 
an  oil  when  in  use.  Gumming  may  seriously  interfere  with  the  distri- 
bution of  oil  to  bearing  surfaces. 

Acid  Test. — Acid  in  mineral  oils  is  usually  sulphuric,  used  in  the  refin- 
ing process  and  not  entirely  washed  out.  An  acid  content  exceeding 
0.3  per  cent,  is  considered  harmful. 

The  friction  test,  to  determine  the  coefficient  of  friction  is  made  with 
a  machine  specially  devised  for  this  purpose.  It  must  be  made  at  stand- 
ard temperature  and  with  a  definite  method  of  applying  the  lubricant. 

There  is  considerable  variation  in  the  characteristics  of  oils  as  given 
by  different  authorities,  but  a  few  values  are  given  in  Table  18. 


LUBRICATION  109 

TABLE  18 


Kind  of  oil 


Gravity,  deg.  B 


Flash  point 
deg.  F. 


High-pressure  cylinder  oil 26 . 0  to  28 

General  cylinder  oil 24 . 0  to  27 

Heavy  engine  oil j  25 . 5  to  28 

Gas  engine  cylinder  oil 26 . 0 

Automobile  engine  oil j  29 . 5 

Air  compressor  oil j 


550  to  600 
530  to  560 
385  to  410 
410  to  500 

430 
500  to  530 


The  cold  test  is  usually  30  or  32  degrees  F. 

A  turbine  builder  recommends  a  pure  mineral  oil  of  600  degrees  test. 

Charles  G.  Sampson,  Trans.  A.S.M.E.,  vol.  35,  p.  151,  says  that  for 
a  blast  furnace  gas  engine  main  bearing,  crosshead,  crank  pin  and  guides, 
an  oil  having  a  specific  gravity  of  0.888,  and  a  flash  point  of  435  degrees  F. 
gave  good  results.  For  the  cylinders,  when  the  piston  speed  was  600 
ft.  per  min.,  an  oil  having  a  specific  gravity  of  0.902,  and  a  flash  point  of 
380  degrees  F.  was  satisfactory,  but  was  not  at  850  ft.  per  min.;  for  this 
speed  a  specific  gravity  of  0.92  and  flash  point  of  502  degrees  F.  were 
used.  He  states  that  with  the  slower  speed  and  lighter  oil  the  wear  was 
excessive. 

Mathot  says  that  the  flash  point  of  gas  engine  cylinder  oil  is  negli- 
gible; that  it  should  have  a  viscosity  similar  to  bearing  oil;  in  fact  the 
same  oil  is  used  for  bearings  and  cylinder  in  the  best  practice. 

Greases  are  composed  of  oils  and  fats  mixed  with  soap,  forming  a 
more  or.  less  solid  mass.  They  are  used  for  heavy  pressures  or  in  places 
difficult  to  lubricate  with  oil;  or,  where  the  movement  is  comparatively 
small,  as  in  valve  gears.  For  extra  heavy  pressures,  solid  lubricants, 
such  as  soapstone,  mica  or  graphite  are  mixed  with  the  grease. 

Systems. — The  individual  system  of  lubrication  is  where  each  bearing 
is  provided  with  an  oil  cup  or  grease  cup,  with  restricted  feed,  or  with  an 
oil  hole  with  intermittent  feed.  The  latter  is  fortunately  passing  out  of 
date. 

The  forced  feed  or  continuous  system  may  be  direct,  in  which  oil 
under  pressure  of  from  3  to  20  Ib.  per  sq.  in.  is  forced  directly  to  the 
bearings,  returned  through  a  strainer  or  filter — and  sometimes  through  a 
cooler — to  the  pump,  where  it  again  starts  upon  its  circuit;  or  it  may  be  a 
gravity  system,  in  which  the  oil  is  pumped  to  an  elevated  tank  located 
upon  the  wall  or  on  the  engine  frame,  and  flows  by  gravity  to  the  bearings. 
In  this  system  the  filter  may  be  in  the  basement,  on  the  ground  floor, 
or  elevated  with  the  storage  tank,  with  which  it  is  sometimes  integral. 


110 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  forced  feed  system  may  have  restricted  feed,  the  supply  being 
throttled  by  a  valve,  or  the  wearing  parts  may  be  continuously  " flooded" 
with  oil,  with  stream  feed,  being  a  form  of  oil  bath,  and  this  method  is 
gaining  favor  for  heat  engine  lubrication.  The  oil  pump  may  be  a 
plunger  pump  operated  from  some  moving  part  of  the  engine  as  the 
rocker  arm  or  valve  lever,  or  it  may  be  a  centrifugal  pump  (see  Fig.  252, 
Chap.  XIX)  operated  by  some  rotating  part.  With  steam  turbines,  a 
gear  pump  is  often  driven  by  an  extension  of  the  governor  spindle.  In 
large  units,  or  in  a  plant  in  which  there  are  a  number  of  engines  of  tur- 


FIG.  62. — Sight-feed  valve. 

bines,  a  separate  pump  driven  by  a  steam  cylinder,  turbine  or  motor 
supplies  oil  for  the  entire  plant.  Manifold  oilers,  sometimes  having  a 
common  pump  and  sometimes  a  pump  for  each  line  which  is  led  from  it 
to  a  bearing,  are  often  used.  Sometimes  a  separate  compartment  con- 
tains cylinder  oil,  the  manifold  oiler  thus  furnishing  complete  lubrication. 
The  splash  system  of  lubrication,  used  mostly  on  the  smaller  engines, 
consists  of  an  enclosed  crank  case  containing  oil  at  such  a  level  that  the 
crank  or  counterbalance  enters  the  surface  at  each  revolution,  splashing 
the  oil  upon  the  bearing  surfaces.  In  single-acting  engines  of  the  trunk- 


LUBRICATION 


111 


piston  type,  this  also  lubricates  the  piston,  which  is  lubricated  separately 
in  other  systems,  sometimes  with  a  different  grade  of  oil. 

Mat-hot  depreciates  the  splash  system,  in  that  it  gives  impurities  no 
time  to  settle;  they  are  thus  brought  continuously  to  the  wearing  surfaces. 

A  combination  of  the  forced-feed  and  splash  system  is  used  in  some 
instances,  notably  with  automobile  engines.  The  oil  is  forced  into  the 
shaft  bearing  by  a  gear  pump,  from  thence  to  the  crank  pin  through 
holes  in  the  shaft  from  which  it  is  thrown.  It  is  claimed  that  the  rapid 
motion  of  the  cranks  (which  do  not  dip  in  the  oil  in  the  bottom  of  the 


Female  Shank 
~~Mcrfe  Shank 

FIG.  63. — "Clear  vision"  sight-feed  oiler. 

oil  pan)  whip  the  oil  into  a  fine  spray,  providing  ample  lubrication  for  the 
cylinders. 

In  feeding  oil  directly  to  the  bearing,  either  with  an  oil  cup  or  with 
the  forced-feed  system,  a  sight  feed  is  generally  employed;  a  sight-feed 
valve  is  shown  in  Fig.  62.  The  valve  may  be  placed  on  top  of  the  bearing 
or  it  may  be  a  part  of  an  oil  cup.  The  flow  of  oil  in  both  cases  may  be 
regulated  by  a  valve.  A  special  sight-feed  oiler  is  shown  in  Fig.  63. 
This  is  made  by  the  Richardson-Phenix  Co. 

In  some  cases  the  sight-feed  valve  is  one  of  several  in  a  manifold  oiler 
located  on  some  part  of  the  engine,  or  turbine  frame,  with  pipes  leading 
to  the  different  bearings;  or  it  may  be  located  near  the  bearing,  but  in  the 
ICturn  pipe.  In  either  case  it  shows  whether  the  oil  is  flowing. 


112 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


When  the  individual  system  is  used  on  engines  which  are  shut  down 
once  or  twice  a  day,  moving  parts  such  as  crosshead  pins,  crank  pins  and 
eccentrics  may  be  supplied  with  oil  cups,  but  as  these  cannot  be  filled 
while  in  motion,  special  devices  must  be  resorted  to  for  continuous  opera- 
tion. For  the  crosshead,  the  most  common  is  the  wiper  and  wiper  cup, 
shown  in  Fig.  64.  Oil  drops  upon  the  wick  from  a  sight-feed  cup  or 
valve  and  is  wiped  off  into  the  cup  near  the  end  of  the  stroke.  Eccen- 
trics are  also  oiled  in  this  manner.  An  alternative  in  both  cases  is  the 
telescopic  oiler  (also  called  trombone  oiler). 

For  the  crank  pin  a  cen- 
trifugal oiler  is  used.  A  con- 
venient form,  especially  for  the 
individual  system,  is  the  Nugent 
oiler.  The  weight  keeps  the 
cup  in  an  upright  position  at 
the  center  line  of  the  shaft. 

Sometimes  a  stand,  fastened  to  the  floor  or  engine 
frame  replaces  the  hanging  weight,  especially  in  the 
forced-feed  system. 

The  telescope  oiler  is  sometimes  used  for  the  crank 
pin  in  center-crank  engines. 

Many   oiling   devices,    as  well  as  system  layouts, 
may  be  seen  in  the  catalogs  referred  to  at  the  end  of 

FIG.    64. — Crosshead  this  chapter. 

Oil  guards,  enclosing  the  crank  and  connecting 

rod,  and  sometimes  the  eccentrics,  are  often  used,  and  may  be  seen  in 
engine  catalogs. 

All  of  these  appliances  are  also  used  with  the  continuous  system.  In 
some  installations,  oil  cups  are  provided  with  each  bearing,  which,  when 
filled  with  oil  through  the  system  piping,  are  shut  off  by  a  valve;  in  case 
the  oiling  system  is  shut  down  for  cleaning  or  repairs,  the  cup  valves  are 
opened  and  lubrication  provided. 

Grease  Cups. — Two  types  of  grease  cups  are  used,  one  a  screw-feed 
cup,  the  other  an  automatic  cup.  In  the  latter,  when  the  handle  is 
screwed  up  to  the  top  of  the  stem,  the  spring  exerts  pressure  upon  the 
grease,  the  feed  being  regulated  by  a  screw  which  forms  a  plug  cock. 

Self-oiling  bearings  have  an  oil  pocket  in  the  bottom  from  which  oil 
is  carried  to  the  top  of  the  shaft  by  rings  or  chains  which  are  supported 
on  the  shaft.  A  gage  glass  on  the  outside  of  the  bearing  indicates  the 
amount  of  oil  in  the  pocket.  They  are  used  on  the  smaller  steam  turbines 
and  sometimes  in  connection  with  a  forced-feed  system;  they  are  also 


LUBRICATION 


113 


used  on  outer  bearings  of  engines  and  sometimes  for  main  bearings  of 
large  engines.  Giildner  objects  to  them  for  internal-combustion  engines 
as  they  reduce  the  wearing  surface  and  weaken  the  bearing.  Bearings 
are  shown  in  Chaps.  XXIX  and  XXXIII. 


FIG.     65. — Gas    engine    main    bearing 
grooves. 


FIG.  66. — Steam  engine  main  bear- 
ing grooves. 


Clearance. — For  proper  lubrication  there  must  be  some  clearance 
between  journal  and  bearing.  No  hard-and-fast  rule  may  be  made' 
and  standards  differ  with  different  builders.  Goodman  says  that  for 
ordinary  lubrication  the  clearance  should  be  0.001  of  the  journal  diameter, 


FIG.  67. — Crank-  and  crosshead  box  grooves. 

and  slightly  more  for  flooded  bearings.     Martin  says  the  clearance  for 
steam  turbine  bearings  should  be  from  0.006  to  0.008  in. 

Oil  Grooves. — It  is  sometimes  claimed  that  oil  grooves  are  unnecessary, 
and  for  the  uniform  conditions  under  which  most  bearing  tests  are  made, 


FIG.  68. — Crosshead  shoe  grooves. 

this  may  be  true;  but  judging  from  the  practice  of  most  builders,  it  is 
apparent  that  there  is  a  general  feeling  in  favor  of  them.  Their  form  and 
arrangement  have  been  the  subject  of  considerable  discussion,  and  no 
attempt  of  a  settlement  of  the  question  will  be  made  here;  but  a  number 

8 


114  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

of  forms  given  by  leading  authorities  and  used  by  successful  builders 
will  be  given.  The  grooves  are  shown  as  though  laid  out  on  a  flat  surface 
equal  in  length  to  the  projection  of  the  circular  arc  which  they  represent, 
and  of  a  width  equal  to  the  length  of  the  bearing. 

There  is  probably  no  doubt  but  that  bearings  should  be  chamfered 
at  the  edges,  and  the  corners  of  chamfer  and  grooves  rounded  as 
phown  in  Fig.  69. 

Bearing  Metal. — It  has  long  been  considered 
that  proper  lubrication  cannot  be  secured  be- 
tween surfaces  of  the  same  material,  and  while 
this  does  not  hold  for  cylinders  and  valves,  at 
least,  it  is  nevertheless  true  that  in  nearly  all 

Fl°'  69~h^mferSVeS  ***  cases>  the  Journals  and  bearings  of  heat  engines 
are  of  different  materials.  The  journals  are 

practically  always  of  steel,  while  the  bearing  boxes  are  either  of  some 
form  of  bronze,  or  are  lined  with  a  soft  metal  such  as  babbitt  or  some 
other  "  anti-friction "  metal,  the  latter  method  being  predominant. 

From  Law  4  for  lubricated  surfaces,  it  makes  little  difference  what  the 
bearing  metal  is  if  the  bearing  is  amply  lubricated,  as  the  journal  is  not 
in  contact  with  the  bearing;  but  bearings  are  not  always  perfectly  lubri- 
cated, so  the  selection  of  the  material  is  important,  especially  for  heavy 
loads  or  high  speeds. 

For  heavy  loads  subject  to  blows,  bronze  is  usually  employed,  although 
the  crank-pin  boxes  of  most  gas  engines  are  lined  with  babbitt.  While  all 
bearings  should  be  properly  fitted,  this  is  especially  true  of  bronze,  but 
if  carefully  fitted  and  well  lubricated,  it  will  probably  outwear  babbitt. 

Goodman  gives  the  advantages  and  disadvantages  of  soft  white  metal 
for  bearings  as  follows : 

Advantages. 

(a)  The  friction  is  much  lower  than  with  hard  bronzes,  cast  iron,  etc.,  hence  it  is 
less  liable  to  heat. 

(b)  The  wear  is  very  small  indeed  after  the  bearing  has  once  got  well  bedded  (see 
disadvantages). 

(c)  It  rarely  scores  the  shaft,  even  if  the  bearing  heats. 

(d)  It  absorbs  any  grit  that  may  get  into  the  bearing,  instead  of  allowing  it  to  churn 
round  and  round,  and  so  cause  damage. 

Disadvantages. 

(a)  Will  not  stand  the  hammering  action  that  some  shafts  are  subjected  to. 

(6)  The  wear  is  very  rapid  at  first  if  the  shaft  is  at  all  rough;  the  action  resembles 
that  of  a  new  file  on  lead.  At  first  the  file  cuts  rapidly,  but  it  soon  clogs,  and  then 
ceases  to  act  as  a  file. 

(c)  It  is  liable  to  melt  out  if  the  bearing  runs  hot. 

(d)  If  made  of  unsuitable  material  it  is  liable  to  corrode 


LUBRICATION  115 

The  crosshead  pins  of  nearly  all  steam-  and  internal-combustion 
engines,  having  comparatively  small  wearing  motion,  are  provided  with 
bronze  bearings,  as  are  some  small  valve-gear  pins,  but  the  main  and 
outer  bearings,  the  crank-pin  box  and  the  crosshead  shoes  are  babbitted. 
Sometimes  steam  engine  pistons  have  rings  or  strips  of  babbitt,  but  this 
is  not  the  rule.  Steam  turbine  shaft  bearings  are  usually  babbitt 
lined. 

The  term  babbitt  is  used  in  a  rather  general  way;  there  are  a  number 
of  anti-friction  metals  on  the  market  which  are  used  for  the  same  purpose. 

The  design  of  the  various  bearings  is  considered  in  connection  with 
the  different  engine  details  of  which  they  form  a  part. 

52.  Bearing  Proportions. — The  bearing  formulas  in  common  use  are 
empirical  and  are  not  based  upon  the  laws  of  friction.  The  most  common 
expression  is : 

PV  =  C  (1) 

in  which  p  is  the  pressure  in  pounds  per  square  inch  of  projected  area, 
V  the  velocity  of  rubbing  surfaces  in  feet  per  minute,  and  C  a  constant 
which  differs  for  various  bearings.  The  mean  pressure  is  assumed,  which 
may  equal  the  maximum  in  some  cases.  In  steam-  and  gas-engine  work 
it  is  the  resultant  of  all  forces  acting  upon  the  bearing  surface,  which, 
for  main  and  outer  bearings  includes  weight  of  shaft,  wheel,  crank,  etc., 
pull  of  belt  and  piston  thrust;  the  latter,  including  inertia,  is  explained 
in  Par.  103,  Chap.  XVI. 

The  piston  thrust  is  often  the  chief  factor  in  internal-combustion 
engines,  especially  in  small  and  medium  powers,  and  consists  of  the  mean 
pressure  due  to  gas  pressure  and  inertia  of  reciprocating  parts.  This 
will  be  further  treated  in  Chap.  XVI. 

A  maximum  pressure  pM  is  also  prescribed,  beyond  which  it  is  best 
not  to  go.  Strength  considerations,  the  most  important  from  the  stand- 
point of  safety,  are  discussed  in  the  chapters  dealing  with  the  details 
involved. 

It  is  apparent  that  preliminary  assumptions  must  be  made,  and  the 
results  checked  by  other  limiting  conditions. 

Let  P  be  the  total  mean  load  on  the  bearing,  I  the  length  and  d  the 
diameter  of  the  bearing  in  inches,  and  N  the  r.p.m.  Then : 


and: 

(3) 


116  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Substituting  in  (1)  gives: 


12C  ~  y'T    C  (4) 

Table  19  gives  values  of  pM  and  C  which  have  been  used  in  practice. 

TABLE   19 


Bearing 

PM 

C 

V 

Main  bearing  
Crosshead  pin  

125  to  200 
1200  to  1500 

50,000 

"8 

a 

W 

Crosshead  pin  —  locomotive  
Crank  pin  

3500  to  4800 
1000  to  1200 

200  000 

s 
1 

Crankpin  —  locomotive 

1200  to  1700 

2 

02 

Crosshead  shoe 

35  to  75 

50  000 

Piston  

25 

30000 

£ 

H 

Main  bearing  
Crosshead  —  or  piston  pin. 

350  to  400 
1000  to  1800 

42,000 

,O 

a 

Crank  pin  

1000  to  1700 

90,000 

6 

"Orosshead  shoe  

35  to  45 

a 

Trunk  piston 

21 

The  values  of  C  for  the  steam  engine  were  given  by  James  Christie 
in  the  American  Machinist,  Dec.  15,  1898,  and  have  been  used  as  a  check 
by  the  author  since  that  time.  For  light  loads,  high  speeds  and  efficient 
lubrication,  it  was  stated  that  the  values  may  be  doubled. 

The  values  for  internal-combustion  engines  are  those  given  by 
Giildner,  and  coincide  with  those  given  by  other  authorities.  In  all  cases 
it  is  assumed  that  steel  journals  and  bearings  lined  with  babbitt  are  used. 
Exception  is  made  for  crosshead  boxes — which  are  usually  of  bronze- 
and  crossheads  and  pistons. 

Where  the  force  acting  upon  a  bearing  is  rapidly  alternating,  better 
lubrication  is  possible,  as  time  is  given  for  the  oil  to  flow  between  the 
surfaces  when  the  pressure  is  removed.  This  obtains  to  some  extent  with 
all  cylindrical  bearings  of  a  reciprocating  engine,  but  especially  for  the 
connecting-rod  bearings;  this  accounts  for  the  larger  value  of  pM  and  C 
given  for  crank  and  crosshead  pins.  These  bearings  also  have  better 
air  circulation  due  to  their  motion. 

A  little  computation  will  show  that  under  the  same  conditions  of 
operation  regarding  temperature,  lubrication,  etc.,  Formula  (1)  gives 
absurd  results  for  widely  varying  surface  speeds  unless  C  is  varied  with 
the  speed.  A  formula  credited  to  the  late  Edwin  Reynolds  has  been 


LUBRICATION  117 

used  for  the  main  bearings  of  engines,  and  has  also  been  used  by  the 
author  lor  heavy-duty  mill  bearings;  it  is: 

pVV  =  K  (5) 

the  notation  being  the  same  as  for  (1).  For  extremes  of  speed  the  con- 
stant K  must  vary,  but  a  given  value  covers  a  wider  range  of  conditions 
than  does  C  in  Formula  (1).  Table  20  from:  the  author's  note  book  will 
illustrate. 

TABLE  20 

Service  K 


Main  bearing — horizontal  engine  . 


Main  bearing — vertical  engine  . 

Sugar  mill  gear  drive 

Sugar  mill  roll  shaft 

Freight  car  journal 


3000 


<i 


30 


2500 
5200 
4900 


The  constants  for  the  engine  bearings  are  modified  to  give  a  smaller 
value  for  small  engines,  assuming  less  efficient  lubrication;  this,  however, 
will  probably  not  hold  today  in  many  cases,  and  the  quantity  within  the 
radical  may  be  ignored.  Neglecting  this,  the  same  value  is  used  for  slow 
turning  sugar  mill  gearing  as  for  the  horizontal  engine.  The  sugar  mill 
roll  shaft  works  under  a  steady  hydraulic  load  of  over  300  tons  at  ex- 
tremely low  speed;  the  diameter  is  limited  due  to  the  housing  bolts,  and 
strength  considerations  reduces  its  length.  Another  extreme  is  the 
freight  car  journal,  which  is  of  limited  size,  under  heavy  pressure  and 
fairly  high  speed. 
Substituting  (2)  and  (3)  in  (5)  gives: 

P  IN 


In  solving  directly  for  the  bearing  dimensions  we  may  take  I  =  kd; 
then : 

d  = 

The  ratio  k  will  be  considered  in  connection  with  the  details  concerned. 
In  main  bearings  it  ranges  usually  from  1.5  to  2.  The  smaller  values  are 
most  us.ed  where  a  large  diameter  is  required  for  strength.  A  very  com- 
mon value  for  stationary  steam  engines  of  moderate  size  is  2. 

Steam  Turbine  Bearings. — Martin  gives  the  values  of  C  in  Formula  (1) 
as  from  150,000  to  180,000  for  stationary  turbines  driving*  electrical  ma- 


118 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


chinery,  and  90,000  for  marine  turbines.  A.  G.  Christie,  Trans.  A.S.M.E., 
vol.  34,  p.  435,  gives  a  value  of  p  from  80  to  100  when  V  is  3600.  This 
gives  C  from  288,000  to  360,000,  and  K  from  4800  to  6000. 

Martin  gives  a  formula  due  to  Nicholson,  which,  insofar  as  pressure 
and  speed  are  concerned,  follows  the  laws  of  friction  of  lubricated  bearings 
qualitatively  at  least;  it  is  as  follows: 


This  formula  is  given  here  as  indicating  the  possible  trend  in  the  design  of 
turbine  bearings.  The  value  of  m  is  given  as  40.  According  to  Martin, 
Dr.  Lasche  successfully  ran  a  bearing  10J4  in.  in  diameter  by  4%  in. 
long,  33  ft.  per  second  with  a  pressure  of  167  Ib.  per  sq.  in.  of  pro- 
jected area.  With  the  same  total  load,  taking  m  as  40,  (8)  gives  a 
length  of  1.96,  for  which  the  pressure  is  372  Ib.  To  adapt  (8)  to  Dr. 
Lasche's  bearing,  m  must  equal  18.3. 

A  value  on  m  which  would  border  upon  conservatism  according  to 
present-day  practice  in  the  United  States  might  be  used  without  destroy- 
ing the  general  rational  form  of  the  equation;  this  would  give  k,  the  ratio 
of  length  to  diameter  between  about  1.75  and  3,  the  smaller  ratio  being 
for  the  higher  speeds. 

To  better  make  a  comparison  of  (4),  (6)  and  (8)  as  applied  to  steam 
turbine  bearings  with  the  value  of  the  constants  already  mentioned, 
approximate  data  from  an  actual  turbine  will  be  taken  and  the  results 
given  in  Table  21. 

P  =  25,000,  N  =  750,  d  =  12  and  V  =2350. 

TABLE  21 


Line 

Formula 

p 

fi 

K 

TO 

i 

k 

1 

Actual 

53.5 

126,000 

2,600 

5.45 

39.00 

3.25 

2 

(4) 

64.0 

150,000 

.... 

32.60 

2.72 

3 

(4) 

76.8 

180,000 

27.20 

2.27 

4 

(4) 

123.0 

288,000 

.... 



17.00 

1.42 

5 

(4) 

153.0 

360,000 

13.60 

1.14 

6 

(6) 

51.2 



2,500 

40.70 

3.40 

7 

(6) 

61  5 

3  000 

34  00 

2  83 

8 

"(6) 

100.0 

4,800 

21.00 

1.75 

9 

(6) 

123  0 

6,000 

17  00 

1.42 

10 

(6) 

87  0 

4,200 

24  00 

2  00 

11 

(8) 

393.0 

40.00 

5.30 

0.44 

12 

(8) 

180.0 



18.30 

11.60 

0.97 

13 

(8) 

98  0 

10  00 

21  25 

1.77 

^•°i 

LUBRICATION  119 

The  diameter  of  the  bearing  may  be  found  tentatively  thus  : 

(9) 


where  H  is  the  horsepower  and  S  the  shearing  stress  (see  Chap.  XXXII). 

It  is  interesting  to  note  that  in  line  6  the  value  of  K  giving  a  length 
about  the  same  as  that  of  the  actual  bearing  (line  1)  is  the  same  as  for  the 
horizontal  engine  and  the  sugar  mill  gears  —  a,  wide  range  indeed. 

Lines  4,  5,  9,  11  and  12  seem  extreme  when  compared  with  present- 
day  practice,  but  2,  3  and  7  are  no  doubt  conservative.  Lines  8,  10  and 
13  give  less  conservative  values,  but  there  is  little  doubt  but  that  friction 
would  b3  less.  Formula  (6)  would  increase  the  length  for  higher  speeds 
and  decrease  it  for  lower  —  a  familiar  practice;  while  (8)  would  do  the  re- 
verse. Formula  (6)  would  therefore  reduce  unit  load  with  increased 
speed,  while  (8)  would  increase  it;  there  are  a  number  of  formulas  given 
by  different  authorities  giving  results  in  a  similar  direction  as  (8),  but  of 
simpler  form  (see  Leutwiler's  Machine  Design).  Formula  (8),  when  m 
is  40,  is  as  exceedingly  generous  when  applied  to  heavily  loaded,  slow- 
speed  shafts,  as  it  is  sparing  with  the  turbine  bearing,  and  gives  absurd 
dimensions.  In  view  of  successful  practice  with  such  bearings,  it  is 
obvious  that  it  is  not  a  general  rational  bearing  formula. 

It  is  apparent  that  all  bearing  formulas  have  their  limits,  but  that  (6) 
will  adapt  itself  to  a  wider  range  of  conditions  with  results  which  are  now 
considered  practical,  than  either  (4)  or  (8),  especially  if  a  single  constant 
is  to  be  used,  although  it  does  not  express  the  laws  of  friction  of  lubricated 
bearings. 

Thrust  bearings  are  used  to  locate  the  shafts  of  steam  turbines.  The 
amount  of  thrust  is  indefinite  and  their  proportions  are  rather  arbitrary. 
In  ship  propulsion,  they  must  take  the  thrust  of  the  propeller;  this  is  due 
to  the  fraction  q  of  the  horsepower  of  the  engine  or  turbine  which  is 
transmitted  to  the  propeller.  In  good  practice,  p  ^  75  for  V  =  500, 
V  being  the  mean  speed  of  the  bearing  surface.  Then  C  =  37,500  and 
K  =  1677.  Let  M  be  the  speed  of  the  ship  in  knots  per  hour,  H  the 
horsepower  of  the  engine  or  turbine  and  a  the  area  of  the  bearing  surface 
in  square  inches.  Then: 

6080  MP 

=  qH 


60  X  33,000 
or: 


M 


120 
Then: 

and : 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


P  = 

= 


aM 


and: 
By  (5): 
and: 


30SqH 

—  ^r~f 

pM 
37,500 


qVH 
~  122M 


1677 

Vv 


a  = 
~ 


(10) 


(ID 


5.08M 


(12) 


Tables  19,  20  and  21  may  be  used  as  a  guide  in  the  design  of  bearings, 

but  it  does  not  seem  wise  to  make 
dogmatic  statements  at  this  stage  of 
bearing  formula  development.  The 
length  of  a  bearing  is  often  com- 
puted, then  modified  by  judgment  or 
by  some  old  rule  of  thumb;  or  it  may 
be  designed  by  rule  of  thumb  and 
checked  by  one  or  more  formulas. 
The  author  has  more  confidence  in 
Formula  (5)  and  the  derived  For- 
mulas (6)  and  (7)  than  in  any 
other  one  formula,  with  the  values 
of  K  given  in  Table  20.  For 
steam  turbines  K  may  be  taken 
from  3000  to  5000  according  to  the 
intrepidity  of  the  designer. 

Eccentric  bearings  have  some- 
times been  used  to  obtain  increased 
bearing  surface  when  space  would  not  allow  an  increase  of  length  except 
by  eccentric  loading.  That  the  desired  end — reducing  bearing  pressure — 
is  often  defeated  by  this  arrangement  may  be  shown  by  the  following 
equations.  Fig.  70  is  a  diagram  of  a  bearing  with  eccentric  load  placed 


FIG.  70. 


LUBRICATION  121 

x  inches  from  the  center  of  the  bearing.  Below  is  a  pressure-distribution 
diagram  which  assumes  a  condition  similar  to  a  rigid  bearing  resting 
upon  an  indefinitely  large  number  of  springs  exactly  alike,  the  deflection 
of  which  measures  the  unit  bearing  pressure  at  that  point  in  the  bearing 
length.  The  actual  case  is  near  enough  to  this  to  give  an  idea  of  the  effect 
of  eccentric  loading. 

The  center  of  area  of  the  load  diagram  being  at  x,  it  remains  to  find  the 
end  ordinates,  pi  and  pz,  which  will  agree  with  this  condition.  For  this 
purpose  the  diagram  may  be  divided  into  two  triangles  whose  centers  of 
gravity  are  1/6  from  the  center  of  the  bearing;  then  equating  moments 
gives  : 


A.lso: 

2l±P.«S=P  (14) 

Rearranging  Equations   (13)   and   (14)   and  adding  together  gives: 


and: 


From  (15)  and  (16)  it  may  be  seen  that  if  x  equals  //6,  p2  is  zero,  while  pi 
is  twice  the  mean  pressure  on  the  bearing. 

53.  Forced-feed  Systems  from  Practice.  —  Few  engine  and  turbine 
builders  manufacture  their  own  lubricating  appliances,  but  these,  which 
are  of  great  variety,  are  obtained  from  manufacturers  making  a  specialty 
of  this  line.  A  few  catalogues  and  bulletins  are  referred  to  at  the  end 
of  this  chapter.  These  may  be  had  for  the  asking,  and  space  will  not 
be  used  to  reproduce  the  numerous  cuts  necessary  for  an  adequate 
description. 

Fig.  7  1  shows  a  Bowser  7F  oil  clarifying  outfit  located  in  the  basement 
of  the  engine  room,  and  connected  with  a  gravity  tank,  forming  a  con- 
tinuous oiling  system.  Oil  is  forced  to  the  tank  by  a  separate  steam 
pump;  it  is  then  led  to  the  bearings,  which  ars  also  shown  fitted  with 
emergency  oil  cups.  The  7F  outfit  is  regarded  as  a  portable  outfit  for 
small  plants,  and  is  given  here  to  illustrate  a  simple  piping  arrangement. 
The  more  complete  Bowser  system  will  be  mentioned  later. 

Taking  a  as  the  area  of  the  wearing  surfaces  of  a  thrust  bearing,  or  the 


122 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


projected  area  of  a  cylindrical  bearing  in  square  inches,  Martin  gives  for 
ihd  heat  generated  by  a  bearing  per  hour  in  B.t.u. : 


, 

h  = 


(17) 


where  V  is  the  velocity  in  feet  per  minute  as  before.  Then  the  weight 
of  oil  which  must  be  supplied  by  the  pumps  in  pounds  per  hour  he  gives 
as  SA/4,  and  the  cooling  surface  in  the  cooler  as  Sft/500  sq.  ft.  He  states 
that  the  tanks  may  have  a  capacity  of  Jfo  the  total  quantity  of  oil 
pumped  per  hour. 


esK  Oil  Supply 


To 
Sewer- 

PIG.  71. — Gravity  system  applied  to  Corliss  engine. 

Filtration  systems  may  be  classified  as  follows: 

1.  Continuous  circulating  systems,  in  which  all  the  oil  used  is  contin- 
uously passed  through  a  filter. 

2.  Partial  filtration,  in  which  part  of  the  dirtiest  oil  is  continuously 
removed  from   the   circulating   system,   passed   through    a   filter  and 
returned  to  the  system. 

3.  Batch  filtration,  in  which  all  of  the  oil  is  removed  and  filtered,  the 
machine  being  supplied  with  fresh  oil  while  the  dirty  oil  is  being  purified. 

Method  (1)  may  be  used  on  steam  and  gas  engines  where  large  quan- 


LUBRICATION  123 

titles  of  oil  are  not  needed  for  cooling  as  in  the  case  of  the  steam  turbine. 
For  the  latter,  the  filtering  surface  required  would  be  excessive,  and  it  is 
unnecessary  to  pass  all  the  oil  through  the  filter. 

Method  (2)  may  be  used  when  excessive  quantities  of  oil  are  used,  as 
for  large  steam  turbine  plants. 

Method  (3)  is  used  for  splash  lubrication  systems  and  where  ring-oiling 
bearings  are  used. 

In  all  clarifying  systems,  precipitation  is  used  to  some  extent,  but  it 
is  a  large  factor  in  Method  (3). 

The  systems  thus  far  illustrated  in  this  paragraph  have  been  continu- 
ous circulating  systems.  Another  example  is  the  2F  filtration  system  of 
the  S.  F.  Bowser  Co.,  Inc.,  Ft.  Wayne,  Ind.  This  system  was  especially 
designed  for  reciprocating-engine  plants  where  basement  space  is  avail- 
able for  the  two  floor  units;  viz.,  the  separator  and  refuse  tank,  and  the 
filter  tank. 

A  quite  generally  applicable  system  is  the  Bowser  6F  filtration  system. 
This  is  a  continuous  system  with  large  precipitating  capacity.  The  system 
was  designed  to  meet  several  important  operating  conditions  as  follows  : 

1.  For  power  plants  where  there  are  no  basements  and  the  engines 
are  located  on  the  ground  floor. 

2.  When  large  volumes  of  water  return  with  the  dirty  oil. 

3.  For  general  application  to  large  plants  requiring  an  individual  filter- 
ing and  circulating  system  of  considerable  capacity,  or  a  central  system 
to  which  all  the  units  in  the  power  plant  are  connected. 

4.  The  6F  system  can  be  used  with  reciprocating  steam  engines,  gas 
or  Diesel  engines,  and  other  machinery  when  a  speedy  separation  of  large 
volumes  of  impurity  is  essential.     This  system  is  of  well  nigh  universal 
application  and  can  be  specified  for  any  class  of  lubrication  requiring 
stream  feed. 

The  6F  system  is  also  recommended  when  it  is  desired  to  filter  a  large 
number  of  batches  so  frequently  that  it  is  necessary  to  pass  oil  through 
with  a  very  short  time  for  precipitation.  It  would  thus  take  the  place 
of  the  5F  system  of  batch  filtration.  A  diagram  of  the  6F  system,  which 
may  aid  in  an  understanding  of  its  operation  is  given  in  Fig.  72. 

54.  Cylinder  Lubrication. — Steam  cylinders  are  lubricated  by  intro- 
ducing oil  with  the  steam,  sometimes  by  means  of  a  simple  gravity  oil 
cup  on  the  steam  chest,  but  in  stationary  engines,  more  often  by  forcing 
the  oil  into  the  steam  pipe  by  a  hydrostatic  lubricator.  A  small  pipe 
from  the  top  of  the  lubricator  connects  with  the  steam  pipe  about  three 
feet  above  where  the  oil  enters  the  steam  pipe;  the  condensed  steam  in 
this  small  pipe  gives  the  head  which  forces  in  the  oil. 


124 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Positive  Lubrication. — Hand  oil  pumps  are  sometimes  used  to  supple- 
ment the  hydrostatic  lubricator,  but  these  depend  upon  the  engine-man  ' 
for  operation.  Probably  the  most  reliable  method  of  lubrication  for 
cylinders,  for  either  steam-  or  internal-combustion  engines  is  by  a  positive 
lubricator  operated  from  some  moving  part  of  the  engine.  The  stroke 
of  the  pump  is  adjustable,  so  that  the  proper  quantity  of  oil  is  fed  at  each 
stroke  of  the  piston.  Such  lubrication  may  feed  into  the  steam  pipe  or 


DIAGRAMMATIC    FLOW    CHART 

SHOWING  COURSE:  or  OIL  THROUGH  THE 

BOWSER    6  F    SY5TCM 


•wwfJTJ* 

RECIEVING    TANK 
FIG.  72. — Diagram  of  Bowser  6F.  filtration  system. 

directly  into  the  cylinder,  and  in  some  large  engines,  may  have  branches 
going  to  the  valves,  especially  if  superheated  steam  is  used. 

In  using  superheated  steam  in  cylinders  previously  using  saturated 
steam,  the  rounding  off  of  the  edges  of  valves  and  ports  has  been  advised 
to  prevent  wiping  off  "the  oil. 

For  high  steam  pressures  and  superheated  steam,  pure  mineral  oil  is 
best,  but  it  is  sometimes  difficult  to  get  it  to  adhere  to  the  surfaces; 
graphite  has  been  found  effective  in  this  case.  It  is  best  mixed  with  oil 
and  may  be  applied  by  a  graphite  lubricator,  a  description  of  which 
may  be  found  in  the  catalogues  cited. 


LUBRICATION 


125 


The  cylinders  of  internal-combustion  engines  are  best  lubricated  by  a 
positive  sight-feed  oil  pump  which  forces  the  oil  onto  the  piston  between 
the  rings  at  the  crank  end  of  the  stroke.  The  usual  place  for  oil  injec- 


FIG.  73. 


FIG.  74.— Lubricating  system  of  Busch-Sulzer  Bros.  Diesel  engine. 

tion  is  between  the  first  and  second  rings  counted  from  the  head  of  the 
piston.  Giildner  says  that  oil  grooves  on  the  surface  of  the  cylinder  are 
of  doubtful  value  if  a  forced  feed  is  not  used  and  the  forcing  of  the  oil  is 


126  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

not  rightly  timed.     He  also  says  that  oil  should  be  supplied  at  several 
points  on  the  circumference  of  the  piston  in  large  engines. 

A  check  valve  is  used  in  the  cylinder  wall  for  forced  feed  and  this  must 
be  constructed  so  as  to  penetrate  the  water  jacket.  Such  a  valve,  fur- 
nished by  the  Richardson-Phenix  Co.  is  shown  in  Fig.  73. 

In  many  internal-combustion  engines,  especially  those  with  small 
cylinders,  the  cylinders  are  not  directly  lubricated;  the  oiling  system 
of  such  engines  properly  comes  under  Par.  53.  In  these  engines,  the 
bearings  have  forced  feed  but  the  cylinders  are  lubricated  by  oil  thrown 
from  the  cranks. 

The  system  of  forced  lubrication  used  on  the  Busch-Sulzer  Bros. 
Diesel  Engine  is  shown  in  Fig.  74.  The  oil  flow  is  indicated  by  the 
arrows. 

References 

Mechanics  Applied  to  Engineering,  Goodman. 

Laws  of  Lubrication  of  Journal  Bearings.    Trans.  A.S.M.E.,  vol.  37,  p.  167. 

Bulletins  of  S.  F.  Bowser  and  Co.,  Inc.,  Fort  Wayne,  Ind. 

Bulletin  of  The  Richardson-Phenix  Co.,  Milwaukee,  Wis. 

Catalog  of  Wm.  W.  Nugent  and  Co.,  Chicago,  111. 


PART  IV— POWER  AND  THRUST 

CHAPTER  XII 

THE  SIMPLE  STEAM  ENGINE 

Notation. 

P  =  pressure  per  square  inch  absolute. 
PB  =  back  pressure  in  pound  per  square  inch  absolute. 
PT  =  terminal  pressure  in  pounds  per  square  inch  absolute. 
PC  =  compression  pressure  in  pounds  per  square  inch  absolute. 
PM  =  mean  effective  pressure  (m.e.p.)  in  pounds  per  square  inch. 
Px  =  maximum  total  unbalanced  pressure  exerted  by  piston. 
p  =  maximum  unbalanced  pressure  per  square  inch.     Also  pres- 
sure used  in  general  discussion. 
p  s  =  maximum  unbalanced  pressure   per  square  inch  taken  as 

some  standard  pressure. 

V  =  volume  of  stroke.     Also  volume  in  general. 
v  =  specific    volume — cubic    feet    per   pound.     Also  volume  in 

general. 

n  =  exponent  in  equation,  pvn  =  constant. 
k  =  ratio  of  clearance  volume  at  one  end  of  cylinder  to  volume 

stroke. 
I  =  ratio  of  portion  of  stroke  up  to  cut-off,  to  entire  stroke; 

commonly  known  as  cut-off. 
x  =  ratio  of  portion  of  stroke  up  to  exhaust  closure,  to  entire 

stroke;  usually  called  compression. 
c  =  ratio  of  volume  of  cushion  steam  in  one  end  of  cylinder  at 

initial  pressure,  to  volume  of  stroke. 
F  =  ratio  of  volume  of  cylinder  feed  in  one  end  of  cylinder  at 

initial  pressure,  to  volume  of  stroke. 
r  =  ratio  of  expansion. 
rc  =  ratio  of  compression. 

C  =  heat  content  of  one  pound  of  steam  (see  Chap.  VII). 
H  =  indicated  horsepower  (i.h.p.). 

A  =  effective  piston  area  in  square  inches  acted  upon  by  the  steam. 
D  =  cylinder  diameter  in  inches. 

127 


128  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Ds  =  diameter  of  cylinder  when  designed  for  standard  unbalanced 

pressure  ps. 

L  =  stroke  of  piston  in  inches. 

N  =  revolutions  of  engine  crank  per  minute  (r.p.m.). 
S  =  mean  piston  speed  in  feet  per  minute. 
W  =  theoretical  steam  consumption  (or  water  rate)  in  pounds  per 

i.h.p.-hr.  measured  from  indicator  diagram. 
w  =  weight  per  cubic  foot  of  dry  saturated  steam. 
/  =  diagram  factor — ratio  of  actual  to  theoretical  m.e.p. 
LogE  =  hyperbolic,  natural  or  Naperian  logarithm;  it  is  the  result 
of  integration  and  used  only  as  a  factor. 

55.  Indicator  Diagrams. — The  indicator  diagram  is  of  great  im- 
portance in  all  phases  of  steam  engine  design,  and  due  to  the  fact  that 
the  maximum  and  minimum  steam  pressures  are  known  with  accuracy 
and  the  opening  and  closing  of  inlet  and  exhaust  valves  may  be  positively 
effected  for  any  load  or  speed,  a  conventional  indicator  diagram  may  be 
safely  used  for  all  design  problems. 

Although  the  curves  employed  are  not  theoretical  curves  for  steam 
from  a  thermodynamic  standpoint,  such  a  diagram  is  commonly  known 
as  a  theoretical  diagram  to  distinguish  it  from  a  diagram  traced  with  an 
indicator,  giving  actual  pressures  within  the  engine  cylinder.  The  differ- 
ence between  the  two  is  treated  under  diagram  factors  in  a  later  paragraph. 

A  conventional  diagram  is  traced  through  a  complete  cycle  of  events  in 
Chap.  Ill,  a  perusal  of  which  is  assumed.  An  important  use  of  the  diagram 
is  the  determination  of  the  m.e.p.,  and  to  greatly  simplify  the  operation, 
admission  and  release  are  assumed  to  occur  at  the  extreme  ends  of  the 
stroke.  The  discrepancy  due  to  this  is  usually  small  except  for  high- 
speed engines  with  single-eccentric  gears,  running  with  light  loads,  and  is 
covered  by  the  diagram  factor  already  alluded  to. 

The  conventional  curve  most  convenient  for  plotting  and  calculation 
has  the  equation: 

pvn  =  constant. 

The  exponent  n  may  equal  or  be  greater  or  less  than  unity.  If  n  is 
unity  the  curve  is  a  rectangular  hyperbola,  which  has  been  most  widely 
used,  and  represents  with  considerable  accuracy  the  expansion  curves 
for  saturated  steam,  and  for  superheated  steam  when  the  superheat  is 
moderate;  this  may  be  seen  in  Fig.  75  which  is  reproduced  from  an  actual 
diagram;  the  hyperbola  is  plotted  from  the  same  point  of  cut-off.  After 
examining  a  number  of  diagrams  from  engines  using  100  degrees  superheat, 
it  seems  that  little  is  to  be  gained  by  using  other  than  the  hyperbola  for 


THE  SIMPLE  STEAM  ENGINE 


129 


the  conventional  curves  of  superheated  steam;  the  deviation  is  probably 
not  greater  than  for  some  saturated  steam  diagrams. 

The  actual  point  of  cut-off  on  many  diagrams  is  not  well  defined,  as 
may  be  seen  in  Fig.  76.     The  A.  S.  M.  E.  Committee  on  Standardizing 
Engine  Tests  (Trans.  A.  S.  M.  E.,  vol.  24,  p.  749),  in  order  that  this 
"  point  may  be  defined  in  exact  terms  for 
commercial    purposes,   as    used  in   steam- 
engine  specifications  and   contracts, "   rec- 
ommends  that  a  horizontal  line  be  drawn 
touching   the  point  of  maximum  pressure 
as  pb,   Fig.   2.      A    hyperbola    from    b  to 
c,  forming  a  continuous  smooth  curve  with 
the  expansion  line  marks  the  cut-off  at  b. 
The  fraction  of  cut-off  is  ab/ag,  and  is  called 

the  commercial  cut-off.  The  actual  cut-off  is  some  later  in  the  stroke 
and  should  be  provided  for  in  valve-gear  design;  the  commercial  cut-off, 
however,  will  be  assumed  in  power  calculations. 

To  draw  the  hyperbola  through  two  points  of  a  curve  on  an  actual 
diagram,  a  clearance  line  must  be  located.  Referring  to  Fig.  77,  let  x 
represent  the  unknown  distance  to  the  clearance  line,  pi  and  pz  the  ab- 
solute pressures  at  the  two  points  and  a  the  horizontal  distance  between 
them.  Then  from  the  equation  of  the  hyperbola: 

from  which: 


FIG.  75. 


Pi 


FIG.  76. 


FIG.  77. 


It  may  also  be  located  graphically  as  shown  by  the  dotted  lines. 
This  line  will  enable  us  to  draw  the  curve,  but  may  not  be  the  actual 
clearance  line. 

The  compression  curve  likewise  fails  to  locate  the  point  of  exhaust, 
and  a  similar  method  may  be  resorted  to  as  shown  in  Fig.  76.  The  con- 
tinuation of  the  compression  curve  up  to  e  gives  the  volume  of  steam  re- 


130 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


tained  in  the  cylinder  by  exhaust  closure,  raised  to  initial  pressure.  This 
is  known  as  cushion  steam.  The  additional  steam  admitted  to  the 
cylinder  up  to  the  point  of  cut-off  is  called  the  cylinder  feed  and  its  volume 
is  given  by  the  line  eb.  The  horizontal  distance  between  the  expansion 
and  compression  curves  at  any  pressure  represents  the  volume  of  cylinder 
feed  at  that  pressure,  as  e'V.  Then  had  the  cylinder  feed,  after  filling 
the  portion  of  the  clearance  space  not  occupied  by  the  cushion  steam, 
been  received  at  maximum  pressure,  the  cut-off  would  have  been  the 
commercial  cut-off  just  described. 

It  is  perhaps  more  satisfactory  for  a  general  non-mathematical  dis- 
cussion to  assume  the  curves  be  and  ed  to  be  actual  steam  curves.  Mathe- 
matical calculations  dealing  with  them  can  be  considered  as  approximate 
only. 

56.  Mean  Effective  Pressure.  —  This  may  be  found  from  an  actual 
diagram  by  dividing  the  area  of  the  diagram  in  square  inches  as  found 
by  a  planimeter,  by  its  length  in  inches,  and  multiplying  by  the  indicator 
spring  scale.  With  the  conventional  diagram  the  mean  ordinate  in 
terms  of  pressure  may  be  found  by  calculation  and  this  process  will  now 
be  followed  step  by  step. 

The  hyperbola,  being 
most  used,  will  alone  be 
considered.  The  area  of 
the  diagram  may  be  best 
determined  by  first  taking 
the  total  area  between 
steam  line,  expansion  curve 
and  the  coordinate  axes, 
then  subtracting  the  por- 
tions which  do  not  belong 

to  the  diagram  proper.  Notation  is  given  at  the  beginning  of  chapter 
and  on  the  diagrams. 

In  Fig.  78,  area  A  =  P\Vi.  From  the  equation  of  the  rectangular 
hyperbola  with  the  asymptotes  as  coordinate  axes: 


pv  = 


or: 
Then: 


p  = 


=  constant 
also         r  = 


-- 

FI 


Area  B 


f 

Jv 


p.dv  — 


log* 


~  =  P1Vl(\ogEV,-\ogEV1) 
=  P1Vl  log*  r. 


THE  SIMPLE  STEAM  ENGINE 


131 


Then: 

Total  area  =  area  A  +  area  B  =  PiFi(l  +  logs  r) 

The  mean  ordinate  in  terms  of  pressure  is: 

_  total  area       PiFi  (  Pi(l  +  logs  r) 

~T7~         ~TT(  ~T~ 

The  back  pressure  of  all  engines  is  greater  than  zero,  and  neglecting 
clearance  and  compression,  is  represented 
by  a  straight  line  of  uniform  pressure  as 
in  Fig.  79;  then: 

Pi(l  +  log*  r) 


if 


(i) 


Because  of  its  simplicity  this  formula 
is  often  used,  r  being  taken  as  the  reciprocal 
of  the  cut-off.  This  leads  to  considerable 
error  with  early  compression. 

It  is  impossible  to  operate  engines  without  clearance,  and  practically 
all  engines  have  a  certain  amount  of  compression;  therefore,  it  is  more 
satisfactory  to  employ  a  conventional  diagram  which  includes  these  items. 
Fig.  80  is  such  a  diagram,  being  a  reproduction  of  Fig.  78  so  far  as  boundary 
lines  and  total  area  are  concerned.  The  areas  A,  B  and  C  are  subtracted 
from  the  total  area,  leaving  the  effective  area.  Substituting  the  new 

' 1 


FIG.  80. 

notation  in  the  formulas  for  ratio  of  expansion  and  total  area  of  Fig.  80 
gives: 

V  +  kV      l+k 


r  = 


IV  +  kV       l  +  k 


(2) 


and: 
Total  area  =  P(IV  +  kV)  +  P(IV  +  kV)\ogE  r  =  PV(l  +  fc)(l+log*r). 


132 
Also: 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Area  A  =  PkV.    Area  C  =  PBxV.   Area  B  =  PckV  log* 


kV 


As  the  curves  are  hyperbolas  it  is  obvious  that : 
PckV  =  PB[(l  -  x)V  +  kV] 


Then: 


AreaB  =  PBV(l  +  k  -  x)  log* 


1  +  k  -  x 
k 


The  ratio  in  this  expression  is  the  ratio  of  compression;  or: 

1+k-x 


rc  = 


The  effective  area  of  the  diagram  then  is: 

E  =  PV[(l  +  fc)(l  +  log*  r)  -  k]  -  PBV[x  +  ATC  log*  rj. 


FIG.  81. 


Then: 


E 


PM  = 

From  (2) : 


Then  (5)  may  be  written : 
•ffc. 


k  = 


log*/)  -  fcJ-PB[x 


a  form  sometimes  convenient. 
From  (3) ,  if  x  is  assumed ; 


(3) 


(4) 


logfi  r)  -k]-  PB[x  +  &rc  log*  rc]        (5) 
1  +  & 


(6) 


(7) 


THE  SIMPLE  STEAM  ENGINE 


133 


Or  if  P.-  is  assumed : 


=  l-k-l=l-  k(rc  - 


(8) 


Fig.  81  is  a  diagram  in  which  the  effective  area  is  shaded.  The  contin- 
uation of  the  compression  curve  up  to  initial  pressure  shows  the  cushion 
steam  cV  and  the  cylinder  feed  FV.  It  may  be  easily  shown  that  the 
product  of  cylinder  feed  and  pressure  is  a  constant  for  a  given  diagram 
when  the  curves  are  hyperbolas,  but  as  this  is  not  a  general  case  it  is 
of  little  importance. 

It  is  often  convenient  to  know  the  cut-off  which  will  produce  a  certain 
m.e.p.  when  other  data  are  known.  This  is  especially  true  when  the 

i.o 


1 

FIG.  82. 


1.0 


engine  is  of  a  type  having  constant  compression.     Formula  (6)  may  be 
transposed  to  read : 

PM  +  Pk  +  Pse 


where 

With  values  of 


r  P(l  +  k) 

6  =  x  +  krc  logs  rc. 

1  +  log£  r 


as  ordinates  and  1/r  as  abscissas,  Fig.  82  has  been  plotted.     In  using  the 
curve  the  ordinate  is  found  from  (9),  the  right-hand  member  of  which 


134 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


contains  only  known  quantities;  referring  to  the  chart,  1/r  is  found,  and 
from  (2) : 

I  =  l(l  +  k)  -  k  (10) 


As  pv  is  constant  for  the  hyperbola  it  is  obvious  from  (2)  that : 

P 


(ID 


The  difference  between  PT  and  PB  is  known  as  terminal  drop.  It  is  clear 
that  the  limiting  value  of  PT  —  PB  for  operating  conditions  is  zero,  and  it 
is  usually  considered  that  this  should  never  be  less  than  the  m.e.p.  which 
would  be  required  to  run  the  engine  without  load,  or  friction  m.e.p. 
This  places  a  maximum  limit  upon  the  ratio  of  expansion.  The  minimum 
limit,  giving  maximum  cut-off,  depends  upon  the  type  of  valve  gear  em- 
ployed, and  if  the  gear  has  no  limit,  upon  the  conditions  of  service  for 
which  'the  engine  is  designed.  The  cut-off  giving  maximum  economy 
is  usually  between  %  and  %,  exceeding  these  limits  in  some  cases.  If 
greatly  exceeded  in  either  direction,  loss  of  economy  ensues,  as  may  be 
seen  from  Fig.  83,  which  is  a  water  rate  curve  of  a  Corliss  engine. 

Assuming  an  economical  cut-off  for  the  rated  power,  the  chart  in 
Fig.  82  may  be  used  to  determine  the  cut-off  required  for  a  certain  per 
cent,  of  overload. 

Example. — Assume  an  initial  pressure  of  125  Ib.  per  sq.  in.  gage  and 

a  back  pressure  of  15  Ib.  absolute,  a  clear- 
ance of  4  per  cent,  and  compression  0.9 
stroke.  For  constant  compression  as  with 
a  Corliss  engine,  find  the  cut-off  for  50  per 
cent,  overload  if  the  rated  cut-off  is  J^ 
stroke  (the  various  ways  of  expressing  the 
ratios — clearance,  compression  and  cut-off 
are  given  here  as  these  are  so  used  in  prac- 
tice; they  must  all  be  reduced  to  fractions  or  decimals  for  numerical 
use  in  the  formulas). 

The  m.e.p.  from  (5)  is: 
PM  =  140[(0.29  X  2.278)  -  0.04]  -  15[0.9  +  (0.04  X  3.5  X  1.2525)] 

=  (140  X  0.62)  -  (15  X  1.075)  =  70.7  Ib. 
For  50  per  cent,  overload : 

PM  =  70.7  X  1.5  =  106  Ib. 
From  (9) : 

1+logr  _  106  +  4.32  +  16.1  _     fiAQ 
~T  "146.5 


I.H.P. 
FIG.  83. 


THE  SIMPLE  STEAM  ENGINE  135 

From  the  chart,  the  corresponding  value  of  1/r  is  0.562,  which  placed 
in  (10)  gives: 

I  =  (0.54  X  1.04)  -  0.04  =  0.522 

This  is  a  longer  cut-off  than  can  be  obtained  under  governor  control 
with  a  single-eccentric  Corliss  gear,  as  will  be  explained  in  Chap.  XX,  so  a 
double-eccentric  gear,  or  some  other  type  must  be  used. 

Uniflow  Engines. — In  applying  the  formulas  of  this  paragraph  to  the 
uniflow  engine,  the  clearance  must  be  determined  which  will  give  a 
compression  pressure  equal  to  the  initial  pressure.  Solving  for  k  in 
(8)  gives: 

1  -  x 

"?:-'  • 

In  determining  k  for  the  condensing  engine,  it  is  well  to  limit  the  value 
of  PB  to  not  much  less  than  6  or  7  Ib.  absolute,  even  though  a  higher 
vacuum  is  to  be  used ;  otherwise,  excessive  compression  pressure  is 
experienced  if  the  vacuum  is  decreased  for  any  reason.  Should  a  better 
vacuum  be  obtained,  the  cut-off  will  shorten  somewhat.  In  determining 
the  maximum  thrust,  the  best  vacuum  (minimum  back  pressure)  should 
be  used. 

57.  Diagram  Factors.  Method  1.— The  A.S.M.E.  Rules  for  Con- 
ducting Steam  Engine  Tests  (Code  of  1902)  contains  the  following: 
"  The  diagram  factor  is  the  proportion  borne  by  the  actual  mean  effective 
pressure  measured  from  the  indicator  diagram  to  that  of  a  diagram  in 
which  the  various  operations  of  admission,  expansion,  release  and  com- 
pression are  carried  on  under  assumed  conditions.  The  factor  recom- 
mended refers  to  an  ideal  diagram  which  represents  the  maximum  power 
obtainable  from  the  steam  accounted  for  by  the  indicator  diagrams  at  the 
point  of  cut-off,  assuming  first  that  the  engine  has  no  clearance;  second, 
that  there  are  no  losses  through  wire-drawing  the  steam  either  during 
the  admission  or  the  release;  third,  that  the  expansion  line  is  a  hyper- 
bolic curve;  and  fourth,  that  the  initial  pressure  is  that  of  the  boiler 
and  the  back  pressure  that  of  the  atmosphere  for  a  noncondensing  en- 
gine, and  of  the  condenser  for  a  condensing  engine. 

"The  diagram  factor  is  useful  for  comparing  the  steam  distribution  losses 
in  different  engines,  and  is  of  special  use  to  the  engine  designer,  for  by  multiplying 
the  mean  effective  pressure  obtained  from  the  assumed  theoretical  diagrams  by  it 
he  will  obtain  the  actual  mean  effective  pressure  that  should  be  developed  in  an 
engine  of  the  type  considered.  The  expansion  and  compression  curves  are  taken 
as  hyperbolas,  because  such  curves  are  ordinarily  used  by  engine  builders  in  their 


136  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

work,  and  a  diagram  based  on  such  curves  will  be  more  useful  to  them  than  one 
where  the  curves  are  constructed  according  to  more  exact  law. 

"In  cases  where  there  is  considerable  loss  of  pressure  between  the  boiler  and 
the  engine,  as  where  steam  is  transmitted  from  a  central  plant  to  a  number  of 
consumers,  the  pressure  of  the  steam  in  the  supply  main  should  be  used  in  place 
of  the  boiler  pressure  in  constructing  the  diagrams."1 

Only  the  first  paragraph  is  given  in  the  revised  code  of  1915,  in  vol. 
37,  but  the  diagrams  are  reproduced  from  vol.  24. 

According  to  this  definition  and  the  accompanying  diagrams,  the 
m.e.p.  of  the  reference  diagram  would  be  due  to  the  expansion  of  the 
cylinder  feed  to  a  volume  equal  to  the  volume  of  stroke,  and  would  be 
calculated  by  Formula  (1),  using  for  the  ratio  of  expansion,  the  value: 

1 


+  k-c 
From  Fig.  82: 

Pc  =  PB(1  +  k  -  x),  or  c  =  ^  (1  +  A;  -  x)  - 
Then: 

r  =  -  -p^  (12) 

*  +  *  -  7?  (1  +  *  ~  *) 

Method  2. — While  the  simple  formula  (1)  is  used  in  the  preceding 
method  to  find  the  m.e.p.  of  the  reference  diagram,  the  value  of  r  given 
by  (12)  requires  as  full  a  knowledge  of  the  valve  setting  as  the  more  exact 
formula  (5),  and  the  labor  of  calculation  is  reduced  but  little.  Further- 
more, Formulas  (1)  and  (12)  do  not  represent  actual  cylinder  operations 
and  there  appears  to  be  little  logical  reason  for  the  assumptions,  as  it  is 
a  physical  impossibility  to  realize  the  maximum  work  from  the  cylinder 
feed  in  this  way.  Neither  does  this  method  bear  any  closer  relationship 
to  efficiency  when  it  is  remembered  that  in  the  use  of  saturated  steam  a 
goodly  percentage  of  the  steam  actually  fed  into  the  cylinder  is  condensed 
by  the  time  cut-off  is  reached,  the  actual  cylinder  feed  being  known  only 
from  an  elaborate  test. 

For  these  reasons  the  author  favors  Fig.  7  as  a  reference  diagram,  and 
Formula  (5)  or  (6)  for  determining  the  m.e.p.  Cut-off  and  compression 
are  assumed  to  be  as  given  under  commercial  cut-off. 

Tables  of  diagram  factors  are  given  in  engineering  handbooks  and 
are  usually  not  the  results  of  recent  practice.  Kent,  Mechanical  Engi- 

1  Trans.  A.S.M.E.,  vol.  24,  p.  753. 


THE  SIMPLE  STEAM  ENGINE  137 

neers'  Pocket  Book,  8th  edition,  p.  931,  following  the  derivation  of  the 
m.e.p.  formula  says:  "The  actual  indicator  diagram  generally  shows  a 
mean  pressure  considerably  less  than  that  due  to  the  initial  pressure  and 
the  rate  of  expansion.  The  causes  of  loss  of  pressure  are:  (1)  Friction 
in  the  stop  valves  and  steam  pipes.  (2)  Friction  or  wire-drawing  of  the 
steam  during  admission  and  cut-off,  due  chiefly  to  defective  valve  gear 
and  contracted  steam  passages,  (3)  Liquifaction  during  expansion. 
(4)  Exhausting  before  the  engine  has  completed  its  stroke.  (5)  Com- 
pression due  to  early  closure  of  exhaust.  (6)  Friction  in  the  exhaust 
ports,  passages  and  pipes. 

"Re-evaporation  during  expansion  of  the  steam  condensed  during  admission, 
and  valve  leakage  after  cut-off,  tend  to  elevate  the  expansion  line  and  increase 
the  mean  pressure. 

"If  the  theoretical  mean  pressure  be  calculated  from  the  initial  pressure  and 
the  rate  of  expansion  on  the  supposition  that  the  expansion  curve  follows 
Mariott's  law,  pv  =  a  constant,  and  the  necessary  corrections  are  made  for 
clearance  and  compression,  the  expected  mean  pressure  in  practice  may  be 
found  by  multiplying  the  calculated  results  by  the  factors  (commonly  called  the 
"diagram  factor")  in  the  following  table,  according  to  Seaton. 

Particulars  of  engine  Factor 

Expansive  engine,  special  valve  gear,  or  with  a  separate  cut-off  valve, 
cylinder  jacketed 0. 94 

Expansive  engine  having  large  ports,  etc.  and  good  ordinary  valves, 
cylinder  jacketed 0.9  to  0.92 

Expansive  engines  with  the  ordinary  valves  and  gear  as  in  general 
practice,  unjacketed 0 . 8  to  0 . 85 

Fast  running  engines  of  the  type  and  design  usually  fitted  in  war 
ships 0.6  to  0.8." 

The  portion  of  the  table  related  to  compound  engines  is  reserved  for 
the  next  chapter. 

Method  3. — A  diagram  giving  the  maximum  possible  m.e.p.  for  steam 
of  any  quality  expanded  to  a  given  terminal  pressure  is  furnished  by  the 
Rankine  cycle  with  incomplete  expansion.  The  m.e.p.  is  given  by 
Formula  (11),  Chap.  VII,  and  is  reproduced  here  with  change  of  sub- 
scripts, as  follows : 

PM  -  5.4  ^—^  +  PT-PS  (13) 

VT  —  0" 

C  and  CT  are  heat  contents  for  initial  and  terminal  pressures  respectively 
and  VT  the  specific  volume  at  pressure  PT.  These  values  may  be  found 
in  Peabody's  entropy  table,  or  calculated  as  explained  in  Chap.  VII.  The 
value  of  <r  may  be  taken  as  0.017  and  is  so  small  that  it  may  be  neglected. 


138  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  effect  of  compression  is  ignored  and  no  idea  given  of  valve  setting 
other  than  that  required  to  give  the  assumed  or  measured  value  of  PT. 

This  method  is  mentioned  more  as  a  matter  of  interest  than  with  any 
idea  of  inviting  its  adoption,  although  it  possesses  advantages  if  maxi- 
mum theoretical  work  in  a  given  cylinder,  with  a  practical  economical 
terminal  pressure  is  desirable  as  a  standard  of  comparison. 

The  m.e.p.'s  of  the  reference  diagrams  for  the  three  methods  outlined 
may  best  be  compared  by  numerical  examples,  assuming  a  certain  valve 
setting. 

Example  1.— Let  P  =  140,  PB  =  15,  I  =  0.25,  k  =  0.05,  x  =  0.85 
and  the  steam  be  dry  saturated  at  point  of  cut-off. 

Method  1.— From  (12),  r  =  3.59  and  from  (1),  PM  =  74. 

Method  2.— From  (2),  r  =  3.5  and  from  (4),  rc  =  4.  Then  from  (5), 
P  M  =70.9. 

Method  3. — From  (11),  PT  =  40.  Then  from  the  entropy  table, 
C  =  1188. 

CT  =  1092  and  VT  =  9.695.     From  (13),  PM  =  78.8. 

Example  2. — Data  the  same  as  before  except  x  =  0.1  as  for  a  uniflow 
engine  and  Pc  =  P. 

Method  1.— From  (8),  k  =  0.108.     Then  r  =  4  and  PM  =  68.6. 

Method  2—k  =  0.108,  r  =  3.09  and  rc  =  9.34.     ThenP^  =  56.1. 

Method  3.— PM  =  78.8  as  before. 

Example  3. — Same  as  example  1  except  that  there  is  no  compression 
and  x  =  1. 

Method  l.—r  =  3.39  and  PM  =  76.7. 

Method  2.--r  =  3.59,  rc  =  1  and  PM  =  73.8. 

Method  3.— PM  =  78.8  as  before. 

There  is  no  doubt  that  in  all  cases  the  results  of  Method  2  are  more 
nearly  like  the  values  from  an  actual  indicator  diagram;  the  diagram 
factor  would  be  nearer  unity  and  less  error  would  follow  if  too  high  a  value 
were  selected. 

58.  Governing. — In  practically  all  stationary  engines  the  speed  is 
constant  except  for  the  slight  variation  necessitated  by  the  practical 
operation  of  the  governor.  Neglecting  this,  the  m.e.p.  is  proportional 
to  the  power  of  the  engine,  and  possible  load  variations  may  be  predeter- 
mined by  the  design  of  the  conventional  indicator  diagram,  which  must 
precede  valve-gear  design. 

Small  engines  are  often  governed  by  throttling  the  steam ;  this  changes 
the  initial  pressure,  the  valve  events  being  the  same  for  all  loads.  The 
conventional  diagram  for  several  different  loads  is  given  in  Fig.  15,  Chap. 


THE  SIMPLE  STEAM  ENGINE  139 

III.  The  maximum  load  is  limited  by  the  maximum  initial  pressure, 
or  pressure  in  the  steam  line;  the  minimum  load  by  the  allowable  terminal 
pressure  PT  discussed  in  Par.  56.  The  total  range  of  load  may  thus  be 
determined  by  finding  the  maximum  and  minimum  m.e.p.,  and  some 
intermediate  load  selected  as  the  rated  load,  which  will  allow  a  certain 
overload.  The  initial  pressure  required  for  the  rated  load  may  be  found 
by  solving  for  P  in  Formula  (5)  ;  or  : 

,     , 


-  k 

As  the  cut-off  is  fixed,  it  is  usually  later  in  the  stroke  than  is  conducive 
to  the  best  economy,  in  order  to  increase  the  maximum  capacity  of  a 
cylinder  of  given  size.  To  have  equal  capacity  with  an  automatic  cut-off 
engine  of  equal  size,  speed  and  pressures,  the  fixed  cut-off  of  a  throttling 
engine  must  be  the  same  as  the  maximum  cut-off  of  the  automatic  engine; 
this  may  be  about  %  stroke.  When  but  small  overload  capacity  is 
required,  a  shorter  cut-off  may  be  used,  resulting  in  better  economy  — 
in  some  cases  exceeding  that  of  the  automatic  cut-off  engine  of  the  same 
power,  especially  at  light  loads. 

The  automatic  cut-off  engine  is  governed  by  changing  the  cut-off. 
In  the  Corliss  engine  and  some  four-valve  engines  with  two  eccentrics, 
the  compression  is  constant,  the  cut-off  only  being  changed  to  suit  the 
load.  This  is  illustrated  in  Fig.  16,  Chap.  III.  The  limits  of  allowable 
load  are  the  maximum  cut-off  and  minimum  terminal  pressure  discussed 
in  Par.  56. 

The  chart  of  Fig.  82  may  then  be  used  to  determine  the  cut-off  for  the 
rated  load  which  will  allow  a  certain  overload;  if  this  cannot  be  fixed 
within  the  economical  limits  —  usually  from  J£  to  J£  cut-off  —  a  compro- 
mise must  be  made  or  a  different  type  of  gear  chosen. 

With  single-valve  engines  and  four-valve  engines  with  a  single  ec- 
centric, the  compression  is  changed  with  the  cut-off;  as  both  affect  the 
area  of  the  diagram  in  the  same  way,  less  change  of  cut-off  is  necessitated 
for  the  same  change  of  load.  A  diagram  for  this  type  is  given  in  Fig.  17, 
Chap.  III.  The  compression  having  been  decided  upon  for  a  given  cut- 
off, it  can  only  be  found  for  another  cut-off  by  the  use  of  a  valve  diagram, 
therefore  the  chart  in  Fig.  82  can  not  be  used  for  determining  the  cut-off 
for  rated  load,  and  it  must  be  found  by  trial  and  error. 

A  combination  of  throttling  and  change  of  cut-off  is  sometimes  used, 
but  as  the  relation  between  cut-off  and  initial  pressure  may  not  be  pre- 
determined except  for  maximum  load,  such  engines  must  be  designed 
for  maximum  power  and  correct  relations  obtained  by  adjustment 


140  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

after  the  engine  is  built.  This  combination  is  sometimes  obtained  unin- 
tentionally, the  throttling  being  due  to  restricted  port  opening  at  short 
cut-off.  This  method  is  shown  in  Fig.  18,  Chap.  III. 

As  more  fully  explained  in  Chap.  XIX,  slight  changes  of  speed  accom- 
pany changes  of  load;  assuming  constant  boiler  and  exhaust  pressures, 
speed  increases  with  decrease  of  load,  and  decreases  with  increase  of  load 
for  both  throttling  and  automatic  cut-off  engines.  This  is  not  on  the 
principle  of  the  work  of  a  horse,  which  can  run  faster  with  a  light  load 
and  must  slow  down  when  given  a  greater  load;  it  is  simply  because  the 
governor  must  run  at  a  given  speed  to  set  the  valve  gear  for  the  cut-off  — 
or  with  the  throttling  engine,  throttle  the  steam  pressure  —  necessary  to 
carry  a  certain  load.  It  is  obvious  that  not  only  change  of  load,  but  any- 
thing which  influences  the  setting  of  the  valve  gear  may  produce  these 
slight  speed  changes;  if  an  engine  changes  while  running,  from  condensing 
to  noncondensing  without  change  of  load,  it  will  slow  down  slightly,  be- 
cause the  back  pressure  has  been  increased,  and  to  keep  the  same  m.e.p. 
the  cut-off  must  be  lengthened  ;  to  effect  the  required  position  of  the  gear 
the  governor  must  run  slower.  In  other  words,  removing  part  of  the 
work-producing  force  is  equivalent  to  increasing  the  load  as  far  as  it 
affects  the  valve-gear  adjustments.  Likewise,  lowering  the  steam  pres- 
sure slows  the  engine  for  the  same  reason  and  not  because  there  is  not 
sufficient  pressure  to  carry  the  load,  provided  the  change  is  not  outside 
the  limit  of  adjustment. 

59.  Piston  Thrust.  —  An  important  consideration  in  engine  design  is 
the  maximum  force  exerted  by  the  piston,  or  piston  thrust.  This  is  the 
product  of  the  maximum  unbalanced  pressure  per  square  inch  p  and  the 
effective  piston  area  A  in  square  inches.  The  unbalanced  pressure  is 
the  difference  between  the  pressure  on  the  two  sides  of  the  piston  and  is 
measured  between  the  steam  or  expansion  line,  which  together  form  the 
forward-pressure  line,  of  one  diagram,  and  the  back-pressure  line  of  the 
other.  This  is  shown  in  Fig.  84,  the  full  lines  representing  the  pressures 
on  opposite  sides  of  the  piston  during  one  stroke,  and  the  dotted  lines 
during  the  other.  Such  a  diagram  is  called  a  stroke  diagram. 
The  maximum  piston  thrust  is  then  : 

Px  =  pA  (15) 

If  D  is  the  diameter  of  the  cylinder  in  inches,  and  the  sectional  area 
of  the  piston  rod  be  neglected  as  being  on  the  safe  side: 


P,  -  P-      =  (P  -  P.-  (16) 


THE  SIMPLE  STEAM  ENGINE 


141 


The  force  required  to  accelerate  the  reciprocating  parts  of  an  engine, 
known  as  the  inertia  of  the  reciprocating  parts,  offsets  a  portion  of  the 
steam  pressure.  This  is  discussed  in  Chap.  XVI  but  is  shown  by  Fig.  85, 
which  is  drawn  by  plotting  the  distance  between  the  two  full  lines  of  Fig. 
84  from  a  horizontal  line.  The  curve  ABC  is  the  inertia  curve,  and  the 
effective  pressures  transmitted  to  engine  parts  are  measured  between 


FIG.  84. 

the  two  lines  as  shown  by  pE.  It  may  be  seen  that  if  cut-off  is  long  enough 
to  lie  over  or  beyond  B,  as  shown  by  the  dotted  line — which  is  usual  in 
practice  for  overloads — the  inertia  has  no  offsetting  effect  upon  the  steam 
pressure,  and  the  maximum  unbalanced  pressure  is  as  given  by  (15) 
and  (16),  or  even  greater. 

The  effect  of  inertia  is  quite  fully  discussed  in  Chaps.  XVI  and  XXI 
(the  last  paragraph  of  the  latter). 


FIG.  85. 

60.  Indicated  Horsepower. — In  Par.  23  of  Chap.  VI,  general  Formula 
(5)  is  derived  for  the  i.h.p.  of  one  working  cylinder  end  of  any  piston  heat 
engine. 

The  formula  is : 

2  X  144     PMvaN 


H  = 


33,000 


ra 


(17) 


142  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

If  L  and  D  are  stroke  and  diameter  of  piston  respectively  : 

LA 

Vs  =  1728 
and  as  m  is  2  for  steam  engines,  (17)  becomes: 


Letting  H  and  C  denote  the  head  and  crank  ends  respectively,  the 
total  power  of  a  single-cylinder  engine  is  : 


H  =  HH+  Ha  =  (PMHAH  +  PMCAc)  (19) 


Formula  (19)  is  general  for  steam  engines;  if  the  engine  is  single-acting 
PMC  is  zero.  In  double-acting  engines  the  piston  rod  reduces  the  piston 
area  at  the  crank  end,  and  if  a  tail  rod  is  used  the  head-end  area  is  also 
reduced.  Formula  (19)  must  be  used  for  determining  the  power  from  a 
test. 

For  the  purpose  of  design  it  is  more  convenient  to  assume  the  areas 
AH  and  AC  equal.  The  piston  rod  may  be  neglected  and  the  discrepancy 
provided  for  by  the  diagram  factor;  or  a  percentage  of  the  piston  area 
may  be  deducted  which  will  be  approximately  correct  for  most  practical 
cases.  In  special  cases  with  exceptionally  large  rods,  more  careful  cal- 
culation must  be  made. 

In  general,  if  dH  is  the  diameter  of  the  tail  rod,  or  extension  for  tan- 
dem engine,  and  dc  is  the  diameter  of  the  main  rod;  and  if  PM  is  the  same 
in  both  ends  of  the  cylinder,  which  is  always  assumed  in  designing,  an 
equivalent  area  which  would  give  the  same  total  power  is  : 

*D  (20) 


Then  (19)  becomes,  for  double-acting  steam  engines: 

w  -  ^.4^.  =  8P**D2LN 
'    198,000    "       252,100 

Piston  speed  is  usually  more  convenient  than  stroke  and  r.p.m  in. 
design.     If  this  is  denoted  by  S: 


S  =  (22) 

•Then  (21)  becomes: 


If  dH  —  dc  -  -R-,  which  is  good  proportion  for  average  practice, 
o 

d  =  0.96  with  tail  rod, 
8  =  0.98  without  tail  rod. 


THE  SIMPLE  STEAM  ENGINE  143 

Assuming  a  mean  of  these  two  values  to  apply  to  double-acting  engines 
with  or  without  tail  rods  with  close  approximation,  and  6  =  1  for  single- 
acting  engines,  special  equations  may  be  written. 
Case  1. — Single-acting,  single-cylinder. 


=    x    M~    ~^      =    *    M~    ~  (<2A\ 

504,200         84,000 
From  which: 

.     D  =  710Vra =  290\S    '  l£2  (25) 

Case  2. — Double-acting,  single-cylinder. 

=  PMD*LN  =  PMD*S 
260,000    "  43,300 
From  which: 


D  -  " = 2°8  (27) 


The  relation  between  S,  L  and  N  may  be  determined  from  (22); 
if  N  is  fixed  by  such  conditions  as  direct-connection  to  an  electric  genera- 
tor, S  or  L  may  be  assumed.  Should  the  ratio  L/D  not  prove  desirable, 
S  and  L  may  be  changed  and  a  new  value  of  D  found. 

If  PM  is  the  theoretical  value  it  should  be  multiplied  by  a  diagram 
factor. 

The  effect  of  piston  speed,  ratio  of  stroke  to  diameter  and  the  factors 
affecting  PM  upon  economy  is  discussed  in  Par.  47,  Chap.  IX.  Their  effect 
upon  capacity  may  be  seen  from  Formulas  (24)  and  (26).  For  a  given 
piston  speed,  long  stroke  means  less  capacity  for  a  given  weight;  for  a 
given  rotative  speed,  long  stroke  means  reduced  cylinder  diameter  and 
piston  thrust,  and  greater  capacity  for  a  given  weight.  Conservative 
designers  prefer  to  keep  the  piston  speed  within  800  ft.  per  min.,  this 
having  been  considered  a  maximum  a  short  time  ago;  but  with  materials 
now  available  there  is  little  doubt  but  that  1000  ft.  may  now  be  considered 
conservative  for  well-constructed  engines  of  good  design.  This  value 
has  been  far  exceeded  in  special  cases  for  stationary  engines,  while  loco- 
motives develop  as  high  as  1700  ft.  per  min. 

Rotative  speeds  vary  greatly  with  size  and  type,  limitations  as  to  the 
latter  applying  generally  to  releasing  valve  gears;  however,  certain 
builders  list  releasing-gear  engines  with  speeds  up  to  200  r.p.m. 

Ratios  of  stroke  to  diameter  vary  for  simple  engines  from  1  to  3,  some- 
times slightly  exceeding  these  limits. 

High  steam  pressure  is  discussed  in  Chap.  IX;  this  is  largely  a  question 
of  boiler  construction  and  operation. 


144  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

From  Formulas  (5)  and  (6)  it  is  apparent  that  a  given  valve  setting  will 
produce  a  certain  theoretical  m.e.p.  Then  from  (26)  it  appears  that  with 
a  given  valve  setting  and  cylinder  diameter  the  i.h.p.  will  increase  with 
the  piston  speed. 

This  is  true  within  certain  limits,  and  the  range  of  speed  over  which 
this  applies  depends  largely  upon  the  capacity  of  steam  passages  and 
ports  to  provide  for  easy  steam  flow;  but  in  any  case,  even  though  there 
were  no  constructional  limitations,  a  very  high  speed  would  result  in 
wire-drawing  through  ports  which  would  decrease  the  m.e.p.,  and  a  point 
would  be  reached  where  the  product  P^S  would  no  longer  increase,  but 
on  the  contrary,  may  decrease.  „ 

Practical  considerations  which  involve  both  port  and  boiler  capacity 
thus  limit  the  horsepower  of  locomotives.  The  tractive  power,  which  is 
the  mean  effort  at  the  rail,  not  including  friction,  is  computed  by  the 
American  Locomotive  Co.  for  a  piston  speed  of  250  ft.  per  min.,  and  a 
mean  cylinder  pressure  85  per  cent,  of  the  boiler  pressure.  For  higher 
speeds  this  is  multiplied  by  speed  factors  less  than  unity.  The  product  of 
speed  factor  and  piston  speed  —  and  therefore  the  horsepower  —  increases 
for  saturated-steam  locomotives,  up  to  700  ft.  piston  speed,  remaining 
constant  to  1000  ft.,  after  which  it  gradually  decreases.  For  super- 
heated-steam  locomotives  the  maximum  is  reached  at  1000  ft.  and  main- 
tained up  to  1600  ft.,  which  is  as  high  as  the  table  goes.  The  maximum 
i.h.p.  for  saturated  steam  locomotives  is  : 


H  =  0. 
and  for  superheated  steam: 

H  =  0.0229PiA 

where  PI  is  boiler  pressure  by  gage  and  A  the  area  of  one  piston  in  square 
inches. 

At  1600  ft.  piston  speed  the  superheated  locomotive  has  25  per  cent. 
greater  capacity  than  the  saturated  locomotive,  to  say  nothing  of  the 
increase  in  economy. 

In  designing  stationary  engines,  the  best  guarantee  against  reduction 
of  power  by  wire-drawing  is  to  proportion  the  ports  so  that  excessive 
steam  velocity  is  not  necessary.  This  will  be  discussed  in  Chap.  XX. 

61.  Theoretical  steam  consumption  may  be  computed  from  either 
an  actual  or  theoretical  diagram.  It  is  the  measure  of  the  cylinder 
feed,  neglecting  condensation  and  quality,  although  it  may  be  applied 
in  case  of  initial  superheat.  The  total  weight  of  steam  in  the  cylinder 
is  usually  measured  at  cut-off,  while  the  cushion  steam  is  taken  at  back 
pressure.  Such  a  computation  is  of  little  practical  value  as  it  is  only  a 


THE  SIMPLE  STEAM  ENGINE 


145 


partial  indication  of  economy;  a  correction  factor,  involving  as  it  does 
so  many  variables,  is  also  of  doubtful  value.     However,    theoretical 
steam  consumption  indicates  whether  a  certain  steam  distribution  aids 
or  counteracts  any  measure  for  improvement  of  economy. 
Let  w    =  weight  per  cubic  foot  of  steam  at  initial  pressure. 
WB  =  weight  per  cubic  foot  of  steam  at  back  pressure. 
W  =  weight  of  steam  per  i.h.p.-hr. 
The  weight  of  steam  per  cycle  of  two  strokes  for  one  cylinder  end  is : 

LA 


1728  1728 

As  there  are  6(W  cycles  per  hour,  the  total  weight  per  hour  is : 
LAN 


28.8 


[w(l  +  *).  -  WB(\  +  k  -  x)]. 


Dividing  this  by  the  i.h.p.  of  a  single-acting  engine  given  by  (18)  gives 
the  water  rate  per  i.h.p.-hr. ;  or 


W 


_  WB(I 


(28) 


FIG.  86. 

If  PM  is  for  a  conventional  diagram  it  should  first  be  multiplied  by  a 
diagram  factor. 

62.  Compression. — The  capacity  of  an  engine  cylinder  of  given  size 
is  not  affected  by  clearance  if  the  expansion  of  the  clearance  steam  is 
equal  to  the  unbalanced  work  of  compression,  or  if  the  two  shaded  areas 
of  Fig.  86  are  equal.  The  volume  of  the  stroke  is  unity.  The  lower 
expansion  line  is  without  clearance  while  the  upper  one  is  for  clearance. 
Then,  assuming  expansion  and  compression  to  be  hyperbolic,  the  un- 
balanced work  of  compression  is: 


A  =  Pck  log*  rc  -  PB(rck  -  k) 
=  PBk[l  +  rc(log*  rc  -  1)] 


10 


146  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  work  of  clearance  steam  is: 
B  = 


Equating  and  solving  for  a  term  containing  rc  gives: 

rc(log£  rc  -  1)  +  1  -  ~~  [(l  +  k)  log*  i-±|  -  I  log*  J]        (29) 

The  right-hand  member  of   (29)   contains  only  known  quantities. 
The  corresponding  values  of  rc  may  be  found  from  Fig.  87. 


I 

1 

/ 

/ 

1 

1 

/ 

/ 

/ 

/ 

/ 

/ 

1 

/ 

^ 

/ 

BIE3        4         56783        10 

rc 
FIG.  87. 

Theoretical  indicated  economy  is  the  ratio  of  work  done  to  steam  sup- 
plied, or  more  properly,  to  heat  supplied.  For  given  pressures  and  con- 
ditions of  steam  this  may  be  expressed  thus: 

E  =  ^  (30) 

From  Fig.  86: 

F  =  i  +  fc  _  c  =  i  +  k  (l  -  §rc)  =  I  +  k  (l  -  §)         (31) 


THE  SIMPLE  STEAM  ENGINE 


147 


Some  numerical  examples,  the  results  of  which  are  contained  in  Tables 
22,  23  and  24,  give  some  idea  of  the  way  E  is  affected  by  different  condi- 
tions, but  it  must  be  remembered  that  the  equations  of  this  paragraph, 
as  in  the  one  previous,  only  show  whether  a  certain  cycle  of  valve  events 
tends  for  or  against  economy;  other  features,  mentioned  in  Chap  IX,  may 
more  than  offset  the  effects  apparent  from  these  equations.  Table  22 
gives  values  of  rc  from  Formula  (29)  and  the  corresponding  values  of  E 
for  three  cut-offs;  also  E  for  the  imaginary  condition  of  zero  clearance, 
and  the  ratio  of  compression  to  initial  pressures.  The  economy  for  the 
clearance  assumed  (4  per  cent.)  is  95  per  cent,  of  that  for  zero  clearance 
in  all  cases.  The  work,  of  course,  is  ijie  same. 

If  cut-off  were  full  stroke  when  k  is  zero,  E  would  be  125,  which,  com- 
pared with  Table  23,  incidentally  shows  the  advantage  of  using  steam 
expansively. 

Table  23  gives  E  for  4  per  cent.,  8  per  cent,  and  for  zero  clearances 
and  (for  the  former  two)  rc  from  Formula  (29) ;  also  for  the  case  of  no 
compression,  or  rc  =  1,  and  when  compression  is  carried  to  the  initial 
pressure.  When  rc  is  calculated  from  (29),  giving  the  same  work  as 

TABLE  22 


P=140       PB  =  IO 

I 

k  =  0.04,  and  r    from  (29) 

k  =  0 

PC/P 

rc 

E 

0.2 

6.70 

275 

290 

0.717 

0.3 

6.36 

245 

258 

0.574 

0.4 

3.95 

218 

230 

0.423 

TABLE  23 


P=140 


0.04 


0.08 


rc  from  (29) 

245.00 

232.00 

rc  =  1 

239.00 

222.00 

258 

rc  =  P/PB 

242.50 

224.00 

zero  clearance,  E  is  the  greatest,  and  from  calculations  with  rc  a  little 
on  either  side  of  those  given  by  (29)  it  is  apparent  that  the  results  of  this 
equation  give  conditions  of  maximum  economy  for  a  given  clearance; 
also,  when  compression  is  thus  determined,  clearance  has  less  effect 


148 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


upon  economy;  i.e.,  for  this  case,  the  8  per  cent,  clearance  gives  95  per 
cent,  of  the  economy  of  the  4  per  cent,  clearance  and  with  no  compression 
this  ratio  is  93  per  cent.;  when  the  compression  is  carried  to  initial  pres- 
sure the  ratio  is  90  per  cent.,  which  points  to  certain  advantages  of  the 
uniflow  principle  which  enable  it  to  overcome  this  apparent  drawback. 
However,  with  shorter  cut-offs,  such  as  obtained  for  rated  load  for  the 
uniflow,  the  high  compression  is  not  so  far  from  that  given  by  (29); 
nevertheless,  the  equation  indicates  that  if  the  clearance  can  be  reduced 
without  counteracting  other  economical  features,  a  gain  in  economy  might 
be  expected. 

In  Chap.  IX  certain  tests  were  mentioned  in  which  economy  was  im- 
proved by  carrying  the  compression  pressure  to  45  per  cent,  of  the  initial 
pressure.  From  Table  22  it  may  be  seen  that  (29)  gives  this  ratio  for  a 
cut-off  some  less  than  0.4  stroke  when  the  clearance  is  4  per  cent. 

TABLE  24 


P=140                          ^5=15                            fc  =  0.06             PC  =  P  when  £  =  0.05 

I 

From  (29) 

By  valve  gear 

X 

rc 

X 

rc 

0.1 

0.53 

8.80 

0.56 

8.22 

0.2 

0.66 

6.56 

0.64 

6.90 

0.3 

0.75 

5.17 

0.72 

5.66 

0.4 

0.80 

4.25 

0.77 

4.83 

0.5 

0.85 

3.45 

0.82 

4.08 

0.6 

0.89 

2.85 

0.86 

3.33 

0.7 

0.94 

2.00 

0.87 

2.90 

Table  24  gives  the  theoretical  values  of  rc  and  x  for  a  single- valve  en- 
gine with  cut-offs  ranging  from  0.1  to  0.7  stroke;  also  in  actual  values 
produced  by  the  valve  gear  for  the  conditions  named  in  the  table,  as- 
suming a  shifting  eccentric  with  constant  lead  and  neglecting  the  angular- 
ity of  the  connecting  rod.  The  method  of  determining  these  values  is 
given  in  Chap.  XX. 

The  value  of  E  varies  but  little  for  quite  considerable  deviations  of 
rC)  so  that  it  appears  from  Table  24  that  a  single-valve  engine  or  its 
equivalent  in  steam  distribution,  provides  for  the  maximum  value  of  E 
for  a  given  cut-off,  if  not  offset  by  other  conditions  mentioned  in  Chap.  IX; 
however,  the  fact  that  in  Table  23  the  maximum  deviation  of  E  from  the 
value  given  by  (29)  is  only  2.5  per  cent,  when  the  clearance  is  4  per  cent., 
and  but  4,2  per  cent,  for  8  per  cent,  clearance,  may  account  for  the 


THE  SIMPLE  STEAM  ENGINE  149 

superior  economy  of  certain  types  of  engines  with  a  constant  compression 
which  may  not  agree  with  that  given  by  (29)  for  loads  usually  carried  by 
the  engine. 

The  indications  of  this  paragraph  are  probably  more  nearly  fulfilled 
in  large  cylinders,  or  when  steam  jackets  or  superheated  steam  is  used. 
In  small  cylinders  the  larger  ratio  of  surface  to  volume  makes  high  com- 
pression less  desirable,  especially  if  the  engine  is  not  of  high  rotative 
speed. 

63.  Standard  Engines. — Most  steam-engine  parts  may  be  standard- 
ized to  the  great  advantage  of  both  the  engineering  department  and 
shops.  This  may  be  done  by  designing  a  line  of  engines  to  carry  some 
standard  maximum  unbalanced  pressure  per  square  inch  which  will  be 
denoted  by  ps.  This  pressure  acting  upon  the  piston  of  a  standard 
cylinder  of  diameter  Ds  will  give  the  same  maximum  thrust  upon  the 
piston  rod  as  some  actual  pressure  p  upon  the  piston  rod  of  a  cylinder  of 
actual  diameter  D.  This  thrust  is  given  by  (16). 
Equating  gives : 

D  TrIV          TrD2 

P.  =  P*-J-   =  P-±- 

From  this: 

Ds  =  D^£  =  KD  (32) 

Then  all  engine  parts  excepting  those  pertaining  directly  to  the  cylinder 
would  be  the  same  as  for  the  standard  engine  with  a  cylinder  diameter  of 
Ds.  The  end  of  the  frame  pattern  containing  the  flange  for  attachment 
to  the  cylinder  may  be  a  loose  piece,  so  that  the  frame  may  be  attached 
to  cylinders  of  different  diameter  by  using  other  flange  pieces. 

To  illustrate,  let  it  be  desired  to  determine  dimensions  of  engine  parts 
for  a  cylinder  24  in.  in  diameter,  carrying  a  pressure  of  150  Ib.  gage  and 
exhausting  to  atmosphere.  Then  p  is  150.  Let  the  standard  pressure 
be  125  (or  ps  =  125).  Then  from  (32) : 


Ds  =  24  X  ^l~^  =  26.3  in. 

When  the  standard  cylinders  are  not  in  fractions  of  an  inch,  it  is  safer 
to  select  the  next  largest  whole  number;  this  would  give  Ds  as  27  inches. 
Should  the  standards  be  based  upon  diameters  which  are  multiples  of  2, 
the  next  larger  is  28  inches,  but  it  is  probable  that  26  inches  might  be 
used  in -this  case,  the  usual  factor  of  safety  taking  care  of  the  slight  dis- 
crepany.  Then  for  the  24-in.  engine  for  a  pressure  of  150  Ib.,  the  parts 
would  be  the  same  as  for  the  standard  26-in.  engine. 


150  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Pins,  rods,  etc.,  may  be  designed  and  tabulated  for  the  full  line  of  cylin- 
ders varying  by  1  or  2  inches,  but  the  frame  pattern  may  cover  a  number 
of  sizes  when  the  advantage  of  reducing  the  number  of  patterns  outweighs 
the  cost  of  using  a  heavier  frame  than  is  necessary  for  the  smaller  engines 
using  the  same  pattern.  In  this  case  the  frame  must  be  designed  for  the 
largest  standard  cylinder  with  which  it  is  to  be  used.  The  diameter  of 
the  bearings  may  be  varied  by  the  box  patterns,  and  the  length,  by  vary- 
ing the  boss  on  the  wheel  side  of  the  bearing;  this  is  sometimes  necessary 
even  if  the  pressure  is  not  changed,  as  may  be  seen  from  Chap.  XXVIII. 

The  same  crank  and  crosshead  patterns  may  also  be  used  for  the 
range  of  sizes  covered  by  the  frame.  The  smaller  this  range,  the  better. 

64.  Application  of  the  formulas  derived  in  this  chapter  will  be  made 
in  the  following  example.  In  such  work  it  is  sufficiently  accurate  to  take 
15  Ib.  as  atmospheric  pressure. 

Example. — Design  a  simple,  noncondensing  Corliss  engine  to  develop 
450  i.h.p.  with  an  initial  gage  pressure  of  125  Ib.  and  a  piston  speed  of  800 
ft.  per  min.  Let  the  cut-off  be  Y±  stroke,  the  compression  0.8  stroke  and 
and  assume  the  clearance  to  be  4  per  cent.  Let  the  diagram  factor  be 
0.9. 

From  (5) : 

P*  =  140[(0.29X2.278)  -0.04] -15[0.8+ (0.04X6X1.79)]  =  68.4  Ib. 
Let  70  per  cent,  overload  be  provided  for;  then: 
PM  =  1.7  x  68.4  =  116  Ib. 
From  (9) : 

1  +  log*  r      116  +  4.32  +  18.4  _  . 
~T  145.5 

From  the  chart,  Fig.  82,  1/r  is  0.71;  then  I  is  0.7,  and  a  long-range  cut-off 
must  be  used  for  such  an  overload.  The  limit  of  cut-off  and  corres- 
ponding overload  for  a  single-eccentric  Corliss  engine  will  be  explained  in 
Chap.  XX. 

From  (27),  including  the  diagram  factor: 


D  -  2°4) .9  X  SI  X  800  "  19"85'  Say  2°  in' 

A  good  ratio  of  L  to  D  is  obtained  if  L  is  48;  then  from  (22)  N  is  100, 
and  the  size  of  the  engine  is: 

20"  X  48"  -  100. 

This  may  be  taken  as  one  of  a  series  of  standard  engines  designed  for  a 
standard  unbalanced  pressure  of  125  Ib. 


THE  SIMPLE  STEAM  ENGINE 


151 


In  tabulating  engines  for  power  it  is  convenient  to  omit  the  diagram 
factor,  as  this  may  vary  for  different  conditions.  Initial  pressure  is 
usually  taken  as  boiler  pressure,  but  if  a  long  steam  line  is  used  there  will 
be  a  pressure  drop  and  the  diagram  factor  will.be  less.  Taking  Pm  as 
the  theoretical  m.e.p.  in  (26),  we  may  write,  where  /is  the  diagram  factor: 

H 


_ 

~       " 


"  43,300 

where  H  is  the  actual  required  i.h.p.  and  HT  the  tabular  value;  the  dia- 
gram factor  may  be  chosen  to  suit  any  particular  case. 

If  125  Ib.  is  taken  as  standard  unbalanced  pressure,  the  value  of  K 
in  (32)  may  be  tabulated  for  other  pressures  as  in  Table  25. 

TABLE  25 


p 
K 

100 
0.895 

110 

0.938 

120 

0.980 

130 

1.02 

140 
1.06 

150 
1.10 

160 
1.13 

170 

1.17 

180 
1.20 

190 
1.24 

200 
1.27 

CHAPTER  XIII 

THE  COMPOUND  STEAM  ENGINE 

Notation. 

P  =  absolute  pressure  in  pounds  per  square  inch. 
PH  =  m.e.p.  in  high-pressure  cylinder. 
PL  =  m.e.p.  in  low-pressure  cylinder. 

PK  =  an  arbitrary  pressure  used  in  Formula  (10) ;  value  from  3  to  6. 
Px  =  maximum  total  unbalanced  pressure  transmitted  by  piston 

rod  to  engine  parts. 
p  =  terminal  pressure  in  pounds  per  square  inch,  absolute.     Also 

pressure  in  general. 

ps  =  standard  unbalanced  pressure  per  square  inch  used  in  design- 
ing standard  simple  engines. 
V  =  volume  of  stroke  used  on  diagrams. 
v  =  volume  used  in  general  discussion. 
d  =  terminal  drop  in  pounds  per  square  inch. 
k  =  ratio  of  clearance  in  one  end  of  cylinder  to  volume  of  stroke. 
I  =  ratio  of  stroke  up  to  cut-off  to  entire  stroke. 
x  =  ratio  of  stroke  up  to  exhaust  closure  to  entire  stroke. 
c  =  ratio  of  cushion  steam  in  one  end  of  cylinder  at  initial  pressure 

to  volume  of  stroke. 

a  =  difference  in  volume  of  low-pressure  and  high-pressure  cush- 
ion steam  at  receiver  pressure. 
/  =  diagram  factor  (see  Par.  57,  Chap.  XII). 
m  =  ratio  of  maximum  absolute  pressure  in  low-pressure  cylinder 

to  pressure  at  cut-off. 
q  =  ratio  of  receiver  volume  to  volume  of  stroke  of  low-pressure 

cylinder. 
R  =  cylinder  ratio — ratio  of  low-pressure  to  high-pressure  volume 

of  stroke. 

rT  =  total  ratio  of  expansion  of  cylinder  feed. 
r  =  ratio  of  expansion  in  one  cylinder. 
rc  —  ratio  of  compression  in  one  cylinder. 
H  =  i.h.p. 

D  =  cylinder  diameter  in  inches. 

Ds  =  diameter  of  standard  simple  engine  cylinder,   which  with 

152 


THE  COMPOUND  STEAM  ENGINE  153 

pressure  ps  will  give  the  same  maximum  thrust  on  piston 
rod  as  compound  engine. 
S  =  mean  piston  speed  in  feet  per  minute. 
W  =  theoretical  water  rate — pounds  per  h.p.-hr. 
w  =  weight  per  cubic  foot  of  steam. 

=  hyperbolic,  natural  or  Naperian  logarithm. 
Subscripts  H  and  1  refer  to  high-pressure  cylinder  while  L  and  2  refer 
to  low-pressure  cylinder.     For  triple-expansion  engines,  1  and  2  refer 
to  intermediate  cylinder  while  L  and  3  refer  to  low-pressure  (see  diagrams 
for  notation). 

65.  Indicator  Diagrams. — Generally  speaking,  the  economy  of  the 
compound  engine  is  so  much  superior  to  that  of  the  simple  engine  that  it 
is  used  in  most  important  engine  installations  notwithstanding  its  greatly 
increased  first  cost.  A  description  of  the  compound  engine,  with  a 
statement  of  the  general  principle  of  operation  is  given  in  Par.  3,  Chap.  Ill, 
while  a  discussion  of  its  economical  advantages  is  given  in  Par.  47,  Chap. 
IX;  these  may  be  considered  as  introductory  to  this  chapter,  although  in 
some  measure  dependent  upon  a  knowledge  of  a  portion  of  its  contents. 
A  simple  but  practical  method  of  determining  pressure  and  volume 
relations  will  be  given  first,  based  upon  the  following  assumptions : 

(1)  That  the  expansion  and  compression  curves  are  hyperbolas. 

(2)  That  the  work  is  equally  divided  between  the  high-pressure  and 
low-pressure  cylinders. 

(3)  That  cut-off,  compression  and  clearance  are  the  same  in  both 
cylinders. 

(4)  That  the  receiver  vol- 
ume is  so  large  that  change  of 
pressure  due  to  exhaust  from 
the  high-pressure  cylinder  and 
intake  of  steam  of  the  low- 
pressure     cylinder     may    be 
neglected,    these   lines    being 
straight  horizontal  lines  which 

coincide.  FIG.  88. 

With  diagrams  and  for- 
mulas established  upon  this  basis,  practical  modifications  may  be  made. 
It  is  of  course  assumed  that  a  condition  of  equilibrium  obtains  between 
cylinder  and  receiver  pressures,  the  effect  upon  these  due  to  starting  and 
change  of  load  being  considered  later.  Conventional  diagrams  for  the 
foregoing  conditions  may  be  arrived  at  as  follows: 

If  all  pressures  of  diagram  1,  Fig.  88  are  multiplied  by  a  constant  and 


154 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


the  corresponding  volumes  divided  by  the  same  constant,  and  the  values 
plotted  to  the  same  scale,  diagram  2  would  be  produced  with  an  area 
equal  to  that  of  diagram  1.  If  this  constant  be  so  chosen  that  the  lower 
boundary  line  of  diagram  1  coincides  with  the  upper  line  of  diagram  2, 
the  required  conventional  diagram  will  be  formed,  as  shown  in  Fig.  89. 
The  cylinder  ratio,  or  ratio  of  volume  of  stroke  of  low-pressure  to  that 
of  the  high-pressure  cylinder  is : 

7?        y« 

= 


Jt  is  obvious  that  R  is  the  constant  previously  referred  to,  and  that: 

'       S   S=p*     :      ";:;;.  (1) 

^i 


T" 


Li 


;    S 

~^=^EEE3~^ 


and: 


FIG.  89. 


?  =  P  1 

R          3       Rz 


From  (2)  : 


(2) 


(3) 


The  receiver  pressure  is  given  by  (1),  after  having  found  the  cylinder 
ratio  from  (3).  The  total  ratio  of  expansion  is  found  from  the  pressure 
ratio  (for  the  hyperbola),  p2  being  usually  assumed  in  compound  engine 
design;  then: 

rT  =  ?1  =  KP*  =  Rr  (4) 

PZ         Pz 


THE  COMPOUND  STEAM  ENGINE  155 

From  which: 

r  =  I  (5) 

and  from  (10),  Chap.  XII: 

I-  ^  -  *  ;          |       (6) 

After  assuming  the  value  of  x,  the  m.e.p.  of  the  high-pressure  cylinder 


PH  =  Pi[^jr?  (1  +  log*  r)  -  fc]  -  P2  [s  +  fc/c  log*  rc]          (7) 


s: 


The  value  of  rc  may  be  determined  from  (4),  Chap.  XII. 

From  (1)  and  (2),  and  the  fact  that  the  quantities  in  brackets  are  the 
same  in  both  high-pressure  and  low-pressure  cylinders,  it  is  clear  that: 

•  Pt  =  if  .     •          (8) 

The  terminal  drop  in  the  high-pressure  cylinder  is  : 

di  =  Rd2  (9) 

66.  Condensing  Engines.  —  For  these  engines  the  value  of  P3  is 
sometimes  so  small  that  the  value  of  R  given  by  (3)  is  excessive,  neces- 
sitating the  selection  of  some  other  value.  To  preserve  uniformity, 
Formula  (3)  may  be  used  by  replacing  P3  by  some  limiting  value,  the 
actual  value  of  P3  to  be  used  should  it  equal  or  be  greater  than  the  limiting 
value,  which  may  be  denoted  by  PK,  and  which  may  have  a  range  of 
from  3  to  6,  according  to  the  judgment  of  the  designer;  then: 


Let  the  ratio  found  by  (3)  be  denoted  by  RK.    Then : 

•  (n) 

Should  PZ,  the  receiver  pressure  be  found  as  in  Par.  65,  using  the  value 
given  by  (10),  it  would  be  too  high;  if  by  (11),  too  low.  If  the  latter 
value  were  used  and  multiplied  by 

3/^K 

V5 

it  will  result  in  nearly  equal  work  in  high-pressure  and  low-pressure 
cylinders;  this  gives  for  the  receiver  pressure: 


156 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


This  is  an  empirical  expression  and  should  be  carefully  checked, 
especially  outside  the  range  of  PK  given  (3  to  6) ;  in  all  cases  it  is  safer 
to  check  the  relation  of  high-pressure  to  lo.w-pressure  work  by  (33). 

If  equal  compressions  are  retained  the  compression  curves  will  not 
lie  on  the  same  hyperbola.  This  affects  the  cut-off  and  terminal  pressure 
in  the  low-pressure  cylinder  if  the  high-pressure  cut-off  remains  the  same, 
and  while  perhaps  not  of  great  importance  numerically,  a  brief  considera- 
tion of  these  relations  may  aid  in  a  better  understanding  of  compound 
engine  operation. 

Going  on  assumptions  1  and  4,  and  approximately  assumption  2  of 
the  previous  paragraph,  Fig.  90  has  been  constructed  for  any  relation  of 


c^ 


QN 


1  fc 


T 

r  # 


«#**- 


l ;^_ ___j— - 


•v7- 


a 


FIG.  90. 


compression  in  the  two  cylinders.     The  volume  of  cushion  steam  in  the 
low-pressure  cylinder  at  pressure  P2  is  c272;  then: 


or: 


rs(l  +  A:2  -  x, 
9.(1  +  fco  —  rc2) 


From  Fig.  90: 


or: 


(13) 


If  a  is  negative  the  low-pressure  curves  lie  to  the  right  of  the  high- 
pressure  curves. 


THE  COMPOUND  STEAM  ENGINE  157 

Cylinder  feed  is  the  distance  between  the  expansion  and  compression 
curves,  and  its  weight  is  assumed  to  be  the  same  in  both  cylinders;  then 
at  receiver  pressure,  a  pressure  common  to  both  cylinders,  the  volume  is 
the  same,  and  the  quantity  aV2  is  also  the  horizontal  distance  between 
the  high-pressure  and  low-pressure  expansion  curves.  Equating  the 
product  of  pressure  and  volume  at  PZ  gives : 

PiV1(l1  +  fci)  =  P*V*(k  +  A*  +  a)  (14) 

To  facilitate  application  a  summary  of  the  foregoing  method  is  given. 
Summary. 

If  P3  ?  PK,     R  =  (15) 


and: 

RK  =  ^  (17) 

Then: 


If  Xi  and  £2  are  assumed  : 


If  p2  is  assumed,  as  is  usual  for  rated  load  : 

-  r2  =  (20) 


and: 


Then  from  (14)  : 

li  =  ^  /2(/,  +  fc2  +  a)  -  fci  (22) 

and: 


Also: 

rT  =  ^  (24) 

If  Zi  is  known,  as  for  an  overload : 

*2  =  J-~-J -•  PJ  -  (fe  +  a)  (25) 


158  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Or,  if  12  is  fixed  as  in  some  engines: 


From  (4),  Chap.  XII: 
and: 


1  +  fei  -  a?i  , 

rci  =  -     —  r—  (27) 


Then  from  (6),  Chap.  XII: 
PH  =  Pi[^*?(l  +  log.ri)  -  A*]  -  P2[zi  +  fcirci  log*  rci]     (29) 


and: 

PL  =  P-rb  +  log*  J-2)  -  fe     -  PBXZ  +  /C2rc2  Iogfirc2       (30) 


Terminal  drop  in  high-pressure  cylinder  is: 

d1  =  ~j-P2  (31) 

In  investigating  an  engine  already  designed  or  built: 


This  assumes  the  stroke  of  high-pressure  and  low-pressure  cylinder  to 
be  the  same  —  which  is  practically  always  true  —  and  may  be  used  in  place 
of  (15)  or  (16),  the  remainder  of  the  calculations  being  as  before. 

Neglecting  the  influence  of  the  receiver,  the  method  of  this  paragraph 
is  general  for  the  assumptions  made.  The  pressure  at  which  high- 
pressure  compression  begins  is  slightly  different  when  the  receiver  is 
considered;  otherwise  all  lines  are  the  same  except  the  back-pressure  line 
of  the  high-pressure  diagram  and  the  steam  line  of  the  low-pressure 
diagram. 

For  preliminary  calculations  for  noncondensing  engines,  and  for  con- 
densing engines  when  P3  is  but  little  different  from  PRy  the  method  of 
Par.  65  may  be  used  and  greatly  simplifies  the  work.  In  fact,  much  less 
refined  methods  than  this  are  often  used  in  practice  —  perhaps  advisedly; 
but  a  better  grasp  of  principles  and  a  sounder  knowledge  of  engine 
operation  are  obtained  by  a  more  exact  analysis,  especially  with  a  wide 
range  of  valve  setting. 

If  the  work  is  equal  in  high-  and  low-pressure  cylinders: 

PHV1  =  PLV,    or,    PH  =  PLR 


THE  COMPOUND  STEAM  ENGINE  159 

The  ratio  of  low-pressure  to  high-pressure  power  is- therefore : 

HL_RP± 

HH  "   PH 

The  statement  that  no  hard-and-fast  rule  can  be  made  for  determining 
cylinder  ratios  is  a  well-known  and  well-worn  one,  and  undoubtedly  true. 
Custom  has  usually  settled  the  question,  and  the  high  ratio  advocates 
have  raised  the  figure  to  a  certain  extent,  but  there  is  still  not  much  uni- 
formity. The  question  is  briefly  discussed  from  a  thermal  standpoint 
in  Chap.  IX,  Par.  47,  and  there  seems  to  be  advantages  in  a  reasonably 
high  ratio. 

A  high  ratio  lengthens  the  cut-off  in  the  high-pressure  cylinder,  and 
for  a  single-eccentric  Corliss  gear — at  one  time  largely  used  on  high-pres- 
sure cylinders  even  after  the  double  eccentric  had  been  used  for  some 
time  on  the  low-pressure  gear — the  overload  capacity  under  governor 
control  is  reduced.  This,  no  doubt,  has  had  some  influence  and  the 
resulting  low  ratio  became  custom. 

With  double-eccentric  Corliss  and  other  long-range  gears,  it  has  some- 
times been  considered  that  a  large  high-pressure  cylinder  was  necessary 
for  large  overload  capacity,  but  Example  1,  Par.  73  shows  that  such  is 
not  the  case,  and  that  practically  100  per  cent,  overload  may  be  carried 
at  J£  cut-off  with  a  cylinder  ratio  of  6.42.  Such  a  ratio  gives  a  good 
cut-off  at  rated  load,  and  in  the  author's  opinion,  has  a  number  of  advan- 
tages. At  any  rate,  why  use  a  large  cylinder  when  a  smaller  one  will 
do  the  work  as  well,  and  with  at  least  as  good  economy? 

67.  Influence  of  the  Receiver. — While  it  may  not  be  considered  of 
great  practical  value  to  treat  the  subject  of  compound  engines  more 
elaborately,  a  better  understanding  of  the  principles  of  compounding  may 
be  gained  thereby:  therefore  sets  of  formulas  will  be  derived  for  four 
cases  which  occur  in  practice  with  2-cylinder,  double-acting  engines  with 
double  expansion,  showing  the  influence  of  the  receiver  on  the  back- 
pressure line  of  the  high-pressure  diagram  and  the  steam  line  of  the  low- 
pressure  diagram.  The  angularity  of  the  connecting  rod  is  neglected  to 
avoid  complicated  calculation;  however,  this  may  be  easily  accounted 
for  by  plotting  the  diagrams  to  a  large  scale  and  measuring  the  volumes 
from  them. 

The  method  employed  is  a  modification  of  that  used  by  Prof.  Unwin 
in  his  Machine  Design,  so  arranged  that  the  valve  events  and  connection 
with  receiver  to  both  ends  of  each  cylinder  may  easily  be  seen  for  the 
entire  cycle.  Equilibrium  of  operation  is  assumed,  the  changes  of  re- 
ceiver pressure  during  starting  and  change  of  load  being  reserved  for  the 


160 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


following  paragraph.  Pressures  and  volumes  are  denoted  by  the  small 
letters  p  and  v,  the  subscripts  referring  to  points  so  numbered  on  the 
indicator  diagrams;  VH,  VL,  CH  and  CL  indicate  volumes  of  stroke  and 
clearance  of  high-  and  low-pressure  cylinders  respectively;  VR  is  the 
receiver  volume. 


Stroke  — -> 
FIG.  91. 


FIG.  92. 


In  numerical  computation,  either  VH  or  VL  may  be  considered  as 
unity,  the  other  values  of  v  being  in  proportion.  For  simplicity,  re- 
lease and  admission  are  assumed  to  occur  at  dead  center,  and  the  expan- 
sion and  compression  curves  are  hyperbolas. 

The  diagram  employed  to  show  the  relation  of  events  consists  of  a  curve 
of  piston  displacement  plotted  on  a  rectified  crank  circle,  as  in  Fig.  91. 


FIG.  93. 

If  angularity  of  connecting  rod  is  to  be  taken  into  account,  this  may 
be  done  as  in  Fig.  92.  This  is  neglected  in  the  diagrams  which  follow. 

In  order  to  avoid  confusion,  the  relation  of  cylinder  and  crank  is  the 
same  as  in  Chap.  XX,  the  cylinder  being  at  the  left  of  the  crank  as  shown 
in  Fig.  93.  The  crank  rotates  clockwise  and  crank  positions  are  num- 
bered from  the  head-end  dead  center  of  the  high-pressure  engine.  For  the 


THE  COMPOUND  STEAM  ENGINE 


161 


tandem  compound  this  coincides  with  the  low-pressure  engine  and  may 
be  shown  in  the  form  given  in  Un win's  Machine  Design,  which  is  also  in- 
troductory to  the  method  to  be  employed. 

Tandem  Compound. — Fig.  94  produces  the  head-end  high-pressure 
diagram  and  the  crank-end  low-pressure  diagram.  Dividing  VR  into 
equal  parts  to  represent  a  rectified  crank  circle,  the  receiver  pressure  may 
be  plotted,  representing,  however,  the  pressure  changes  for  but  one  stroke 
of  the  pistons. 


FIG.  94. 

Beginning  with  the  head  end  of  both  diagrams,  operations  may  be 
traced.  From  0  to  1  on  the  high-pressure  diagram,  steam  is  admitted 
to  the  cylinder.  At  1,  expansion  of  all  steam  including  clearance  steam, 
begins  in  the  high-pressure  cylinder;  at  the  same  time  steam  is  being 
forced  out  of  the  low-pressure  cylinder  until  14  is  reached,  when  compres- 
sion begins  and  is  finished  at  15.  At  the  same  time  high-pressure  ex- 
pansion ends  at  2.  High-pressure  release  and  low-pressure  admission 
are  now  assumed  to  occur  simultaneously.  Pressure  pi5  in  the  low-pres- 
sure clearance,  pressure  p2  in  the  receiver  and  pb  in  the  high-pressure 
cylinder  are  now  all  changed  to  ps  (=  pio),  as  both  cylinders  are  now 
open  to  the  receiver. 

The  piston  now  starts  on  the  return  stroke  from  right  to  left,  and  as 
the  low-pressure  piston  displaces  volume  as  it  takes  steam  from  the 
receiver,  more  rapidly  than  the  high-pressure  piston  as  it  exhausts  into 
11 


162 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


the  receiver,  the  pressure  falls  until  low-pressure  cut-off  is  reached  at  11. 
Low-pressure  expansion  now  begins,  and  at  the  corresponding  pqint  4 
of  the  high-pressure  diagram,  compression  of  steam  into  the  receiver  is 
begun  by  the  high-pressure  piston.  This  continues  until  the  high- 
pressure  exhaust  valve  closes  at  5,  shutting  off  the  communication  with 
the  receiver.  Compression  in  the  high-pressure  cylinder  is  complete 
at  6,  and  at  the  same  time  low-pressure  expansion  is  complete. 

Fig.  95  is  a  modification  of  this  diagram,  from  which  indicator  dia- 
grams may  be  traced  for  both  ends  of  each  cylinder.  The  diagrams  used 
in  forming  the  equations  are  in  full  lines.  With  this  method  the  receiver 
volume  may  not  be  drawn  to  scale,  but  all  shaded  areas  extending  into 


FIG.  95. 

the  central  square  indicate  communication  with  the  receiver.  All  such 
areas  join  a  diagonal  line  as  shown,  and  if  shaded  areas  from  two  cylinder 
ends  meet,  they  are  both  in  communication  with  the  receiver.  The  upper 
end  of  the  diagonal  corresponds  to  head-end  dead  center  of  the  high- 
pressure  crank,  and  following  this  line  to  its  lower  extremity,  all  events 
of  the  cycle  may  be  traced. 

The  indicator  diagrams  of  Figs.  95,  97,  100  and  103  are  for  a  small  re- 
ceiver, exaggerating  the  pressure  variation.  The  volume  of  the  low- 
pressure  cylinder  is  drawn  one-half  the  computed  volume  to  save  space. 
No  special  work  division  is  aimed  at,  the  diagrams  being  used  to  illustrate 
the  method.  Figs.  96,  99,  102  and  105  are  for  condensing  engines  with  a 


THE  COMPOUND  STEAM  ENGINE  163 

cylinder  ratio  of  6.42,  a  back  pressure  of  2  Ib.  absolute  and  a  receiver 
volume  equal  to  the  volume  of  the  low-pressure  cylinder. 
From  Fig.  95  the  following  equations  may  be  written. 

p&i  =  P&2     (a)          ptv2  +  p&R  +  pucL  =  Pa(vz  +  VR  +  CL)     (b) 


VR  +  CL)  =  p4(*>4  +  VR  +  Vu)     (c)     p3 
2>4  =  Pn     (e)         p4(t>4  +  VR)  =  pb(v5  +  VR)     (/) 
PbV$  =  P&CH     (g)         pnVn  =  pizVn     (h) 
From  (a),  (6),  (c)  and  (i): 

P*(V4  +   VR  +  Vn)  =     piVi   +  Pl4^14   +  P&R       (j) 

and  from  (/), 


Then  from  0')  and  (fc): 

(I) 


From  (e),  (I)  and  (i),  an  equation  containing  only  known  quantities 
is  derived;  or: 

/«„»•„    _t_   /n.  .«t. , 

-TV  <«) 


The  equations  following  give  all  other  points  on  the  diagram. 
From  (i)  : 


(„)     From  (*),     pu  -  (o) 


From  (/),  P6  =  a     (P)     From  (g),      p6  -  (g) 


From  (a),  p2  =  (r)     From  (c)  and  (d),  pz  =  pw 

V2 


+  Vn)  ,  , 


V*  +  tfc  +  Cx, 
If  ZL  =  low-pressure  cut-off: 

V4  =    »2  -  IL.VH  (t) 

Combined  diagrams  for  the  head  end  are  shown  in  Fig.  96,  in  which 
the  receiver  volume  equals  the  volume  of  the  low-pressure  cylinder. 
The  dotted  lines  show  the  indefinitely  large  receiver  assumed  in  Par.  66. 
Except  for  the  back-pressure  line  of  the  high-pressure  diagram  and  the 
steam  line  of  the  low-pressure  diagram  the  lines  are  practically  identical. 
The  larger  the  receiver  the  more  nearly  will  the  lines  mentioned  approach 
the  dotted  lines, 


164 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


It  is  plain  that  the  influence  of  the  receiver  is  to  increase  the  maximum 
pressure  in  the  low-pressure  cylinder,  increase  the  work  done  in  this 
cylinder  and  decrease  the  work  of  the  high-pressure  cylinder. 


FIG.  96. 


FIG.  97. 


Cross  Compound. — Diagrams    for    cross-compound    engines   will   be 
drawn  with  the  high-pressure  crank  leading  the  low-pressure  crank  by 


THE  COMPOUND  STEAM  ENGINE 


165 


90  degrees.  They  would  be  identical  with  those  in  which  the  low-pressure 
crank  leads  so  long  as  angularity  of  the  connecting  rod  were  neglected. 

There  are  three  sets  of  diagrams  required  to  show  conditions  commonly 
found  in  practice  due  to  the  action  of  the  governor  upon  the  valve  gear. 

Case  1. — In  which  low-pressure  cut-off  occurs  after  compression  in 
one  end  of  the  high-pressure  cylinder  and  before  release  in  the  other  end. 
The  diagram  is  shown  in  Fig.  97  and  is  similar  to  Fig.  94  except  that  the 
piston  displacement  curves  for  the  low-pressure  cylinder  are  not  in  phase 
with  the  high-pressure  curves. 

In  Fig.  98, 


V*    =    I' 5 


FIG.  98. 


and  vn  is  determined  by  the  position  of  the  low-pressure  crank  when  the 
high-pressure  crank  is  in  the  compression  position  as  in  Fig.  98.     Then : 


versin  A  — 


VH 


B  =  90  -  A     and 


VL 


L  •      D    i 

=  -    versm  B  +  CL. 


Or,  if  xn  —  the  fraction  of  stroke  up  to  high-pressure  compression : 

Vn  =    t'zX0.5  -  \/XH  -  XH2)  +  CL. 

If  #11  >  viz,  use  Case  2. 

From  Fig.  97  the  following  equations  may  be  written : 

Pivi  =  pzVz     (a)         p2v2  +  pizVR  =  Pz(v2  +  VR)     (b) 

P3  (02   +   VR)    =    P4(V4   +   VR)  (c) 

Pity*  +  VR)  +  p^CL  =  p5(>4  +  VR  +  CL)  (d) 

Ps(v*  +  VR  +  VL)  =  p6(v&  +  VR  +  Vu)  (e) 

#10    =    Po        (/)  Pll    =    PG        (g)  ?>6»>6    =    P&H        W 

PG(VR  +  Vn)  =  piz(vR  +  0i2)      (i)'         pizVn  =  pnVis     (j) 


166  DESIGN  AND  CONSTRUCTION  OP  HEAT  ENGINES 

From  (a)  to  (e): 

ffl 


FIG.  99. 


FIG.  100. 
From  (i)j  multiplying  both  sides  of  the  equation  by  VR: 


VR 


(m) 


THE  COMPOUND  STEAM  ENGINE 


167 


Substituting  (k)  and  (wi)  in  (I),  solving  for  p6  and  equating  with  (g) 
gives  an  equation  containing  only  known  quantities: 


=  Pn  = 


+.. 


+  VR  + 


(n) 


The  equations  which  follow  locate  all  other  points  on  the  diagram. 
From  (h), 

p7  =  P^?  (0)  From  (z),     pi2  =     6..  g  ,  -  (p) 


CH 


From  (j), 


From  (e)  and  (/), 


From  (k), 
Pefye  +  VR  - 

V4  4-  VR   4 


(r) 

w 


From  (d)  and  (k),     p* 
From  (a),     p2  =  —^ 


v*  +  VR 
(u)         From  (c), 


(0 


Combined  diagrams  for  the  head  end  are  shown  in  Fig.  99,  using  the 
same  data  as  in  Fig.  96.  The  dotted  lines,  as  before,  show  an  indefinitely 
large  receiver,  and  the  variation  of  pressure  and  work  due  to  a  receiver 
of  practical  size  may  be  seen. 

Case  2. — In  which  low-pressure  cut-off  occurs  before  compression  in 
the  high-pressure  cylinder.  This  is  shown  in  Fig.  100,  which  is  in  other 
respects  like  Fig.  97. 

In  Fig.  100, 


168  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and  VQ  is  determined  by  the  position  of  the  high-pressure  crank  when 
the  low-pressure  crank  is  in  the  cut-off  position,  as  in  Fig.  101.     Then: 


versin  B  =  ~^       .  A  =  90  -  B  and  v6  =  ^versin  A  +  ca. 

VL  £ 

Or,  if  1L  =  fraction  of  stroke  up  to  low-pressure  cut-off  : 

V&  =    ^(0.5  —   \^1L  —  1L2)  +  CH. 

If  vG  <  v7}  use  Case  1. 

From  Fig.  100,  the  following  equations  may  be  written: 
(a)         p&z  +  p7vR  =  p3(v2  +  VR)     (b\ 


VB)    = 
VR)    +    pUCL    =    P5(V4    +   VR   +    CL)         (d) 

VR  +  CL)  =  p6(>6  +  VR  +  vn)      (e) 
Pio  =  Pb     (/)          Pii  =  P&     (g)         Pe(v&  +  VR)  =  p7(v7  +  VR)      (h) 
p7v7  =  p8cH     (i)         piivu  =  pizVi*     (j)         puVu  =  pibCL     (k) 

From  (a)  to  (e),  and  (k), 

PG(VG  +  VR  +  Vn)  =  piVi  +  p7vR  +  puvu     (I) 

From  (h),  multiplying  both  terms  of  the  equation  by  VR  and  solving 
for  p7vR, 


,          . 


Substituting  (m)  in  (2),  solving  for  p6  and  equating  with  (g)  gives 
an  equation  containing  only  known  quantities. 


(r,  +  ...  +  «u)  - 


VJ          I         VR 

All  other  points  are  located  by  the  following  equations: 
From  (h),      p7  =  — — ~ — —        (o).  From  (i),     p%  =  -^          (p) 

From  (j),     piz  =       ^n        (#).  From  (k),     pi6  = 

/  \          i    //«\  P&(VQ  ~\~  V R  -f-  fii) 

From  (e)  and  (/),        p5  =  pi0  =    ^ +T~ 


From  (d)  and  (*),       P<  =  p'(^  +  "B  +  ^  ~  P"y"  (0 

f  4  T  v« 

From  (a),        p2  =  *&       („).        From  (c),    p3  =  ^f^-          (») 


THE  COMPOUND  STEAM  ENGINE 


169 


Combined  diagrams  for  the  head  end  are  shown  in  Fig.  102. 
Case  3. — In  which  high-pressure  release  occurs  before  low-pressure 
cut-off,  which  is  greater  than  one-half  stroke.     This  is  shown  in  Fig.  103. 


FIG.  102. 


From  Fig.  103, 


FIG.  103. 


VH    , 
TT  H"  CB 


170 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


and  v4  is  determined  by  the  high-pressure  crank  position  at  the  time  of 
low-pressure  cut-off  as  shown  by  Fig.  104. 
Then: 

2(wi6  -  flu).       D  . 


cos   A  = 


and      vt  =       (1  +  cos  B)  +  e.. 


Or,  if  1L  =  fraction  of  stroke  up  to  low-pressure  cut-off: 

v,  =  1^(0.5  +  VlL  -  k2)  +  Ca. 

.  The  value  of  v\\  is  determined  by  the  low-pressure  crank  position  at 
the  time  of  high-pressure  compression  and  is  found  as  for  Fig.  101,  re- 
placing VQ  by  Vft  or: 

CL 


.p. 


.5  —  V  X     — 


H.P. 


FIG.  104. 


in  which  XH  =  fraction  of  stroke  up  to  compression  in  the  high-pressure 
cylinder. 

From  Fig.  103,  the  following  equations  may  be  written: 


(g) 


PQ 


From  (a)  to  (c), 
From  (j)  to  (Q, 


CL)  = 

=  Pi     0') 
+  ww)         (0 

=  piScL     (n) 
vfl  +  vu)  =  piVi  +  p\*(vR  + 
+  vu)  =  p7(vR  +  vn)     (p) 


(m) 


Substituting  (p)  in  (o)  and  solving  for  p4  gives: 

p*  =  ^7+ 


THE  COMPOUND  STEAM  ENGINE 

From  (/)  to  (h),  and  (ri): 

r  VR  -f  Vn)  — 


171 


v<  t  r,  (r) 

Equating  (q),  (r)  and  (7)  and  solving  for  p7  gives  an  equation  con- 
taining only  known  quantities. 

piVi(v*  +  VR)  +  pnVn(v*  ~\-  VR  - 


(0 

w 


The  following  equations  locate  all  other  points. 

Pl(Vl    +   VR   ~ 


From  (h)  and  (i), 
From  (k),     p8  = 


From  (S)  and  (»), 


?6    =    PlO    = 

V6  T   t'B 

(w).  From  (Z), 


+ 


VR 


FIG.  105. 
=  PeC^s  +  ^g  +  CL)  —  Pi7^n 

Vs  +  VR 

From  (w),   PIS  =  —   -  (a;).     From  (e)  and  (/),   i 
From  (m),  p16  =  Pl4^14 

.     From  (c)  and  (d) ,  p3  =  y 


Pu  = 


(«) 

(») 

(z) 

(B) 


Combined  diagrams  for  the  head  end  are  shown  in  Fig.  105. 

The  foregoing  discussion  covers  all  practical  cases  of  tandem  and  cross- 
compound  engines,  the  latter  with  cranks  90  degrees  apart.  It  is  obvious 
that  the  same  analysis  may  be  applied  to  any  arrangement  of  cylinders 


172  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and  cranks,  to  3-cylinder  compounds,  or  to  the  large  variety  of  cases 
found  in  triple-  and  quadruple-expansion  engines;  but  due  to  the 
large  number  of  formulas  involved  and  their  infrequent  application,  it 
was  thought  best  to  limit  the  treatment  to  the  2-cylinder  compound 
engine. 

The  method  of  this  paragraph  is  helpful  in  the  study  of  compound- 
engine  diagrams,  the  ability  to  rightly  interpret  which  being  impossible 
without  the  knowledge  due  to  such  an  analysis.  For  instance,  the 
sloping  line  of  the  low-pressure  diagram  has  often  been  attributed  to  wire- 
drawing, and  in  one  instance  to  the  author's  knowledge,  the  point  in 
the  steam  line  of  the  low-pressure  diagram  of  a  cross-compound  engine 
caused  by  high-pressure  exhaust  closure  was  thought  by  a  consulting 
engineer  to  be  the  point  of  cut-off. 

The  distorted  diagrams  sometimes  taken  from  triple-expansion  engines 
are  not  the  result  of  improper  valve  setting  or  too  small  ports,  but  are 
due  to  compression  and  expansion  of  receiver  steam. 

The  sharp  corners  do  not  appear  on  actual  diagrams,  and  with  large 
receivers,  both  high-  and  low-pressure  diagrams  often  closely  resemble 
those  from  simple  engines.  If  plotted  to  different  scales  for  combining, 
however,  the  general  form  is  more  like  the  conventional  diagrams. 

68.  Governing. — The  speed  regulation  of  compound  engines  is 
accomplished  in  two  ways:  (1)  by  direct  governor  control  of  both  cylin- 
ders; (2)  by  direct  control  of  the 
high-pressure  cylinder  only.  In 
method  (1)  the  increased  or  de- 
creased weight  of  steam  exhausted 
into  the  receiver  is  provided  for  in 
the  low-pressure  cylinder  by  a 
lengthened  or  shortened  cut-off, 
thus  maintaining  a  nearly  con- 
stant receiver  pressure,  as  shown 


I 


FIG.  106.  in  Figs.  99,  102  and  105,  in  which 

the  receiver  pressure  at  the  time  of 

low-pressure  cut-off  is  the  same.  Assuming  an  indefinitely  large  receiver, 
the  receiver  pressure  would  be  constant  for  this  method  of  governing  if 
the  low-pressure  cut-off  were  rightly  selected,  as  shown  in  Fig.  106. 
Both  cylinders  then  respond  to  the  governor,  and  the  regulation  should 
be  practically  as  sensitive  as  with  a  simple  engine. 

In  method  (2)  the  low-pressure  cut-off  is  fixed,  and  for  different  loads 
on  the  engine,  equilibrium  requires  different  pressures  as  shown  in 
Fig.  107,  in  which,  for  convenience,  the  compression  curves  are  shown  on 


THE  COMPOUND  STEAM  ENGINE 


173 


FIG.  107. 


the  same  hyperbolas  for  all  loads,  necessitating  different  points  of  high- 
pressure  exhaust  closure. 

If  these  receiver  pressures  were  attained  instantly  upon  change  of 
high-pressure  cut-off,  the  control  would  probably  be  as  sensitive  as  with 
method  (1)  and  perhaps  more  sensitive,  as  the  high-pressure  terminal 
drop  being  more  uniform,  the  change  of  total  diagram  area  is  greater 
for  a  given  change  of  high-pressure  cut-off;  but  a  number  of  strokes  are 
required  to  raise  or  lower  the  receiver  pressure  to  meet  the  new  load,  dur- 
ing which  time  the  change  of  high- 
pressure  cut-off  must  be  greater 
than  for  method  (1)  with  a  con- 
sequent increased  speed  fluctu- 
ation. As  the  receiver  pressure 
changes,  the  cut-off  gradually 
changes  to  that  required  for 
equilibrium  at  the  new  load. 
It  is  obvious  that  the  larger  the 
receiver  the  longer  the  time  re- 
quired for  pressure  adjustment, 

which,  however,  means  a  more  uniform  receiver  pressure,  an  advantage 
gained  by  a  large  receiver  in  any  case.  Slightly  better  economy  is 
obtained  by  method  (2)  according  to  some  engine  builders. 

Were  it  not  for  the  great  labor  of  determining  the  changing  high- 
pressure  cut-off  during  receiver-pressure  adjustments  to  correspond  to 
a  new  load,  the  equations  of  the  previous  paragraph  could  be  used  for 
this  purpose.  Some  idea  of  the  pressure  changes  may  be  given  by 
assuming  a  constant  new  cut-off.  Then  taking  the  data  from  which 
Fig.  96  for  a  tandem  compound  is  plotted,  the  pressure  at  low-pressure 
cut-off  is : 

7?4  =  PU  =  20.2  Ib. 

The  maximum  receiver  pressure  is: 

pb=  22.15lb. 

Assuming  the  low-pressure  cut-off  as  given  and  the  high-pressure 
cut-off  increased  to  0.5  stroke,  the  new  value  of  pn  for  equilibrium  is, 
from  (m) : 

p4=  pu=   43.3lb. 

Then  finding  pb  from  (k)  and  p4  again  from  (j)  by  substituting  the 
value  of  p5  just  found,  and  so  on,  alternating  between  (j)  and  (A;),  p4  grad- 
ually rises  and  is  shown  in  Fig.  108,  in  which  ordinates  are  changes  of  pres- 


174 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


sure  from  20.2  to  43.3,  and  abscissas  are  the  number  of  strokes  since  the 
change  of  high-pressure  cut-off.  The  calculations  were  made  with  a 
slide  rule,  but  it  is  obvious  from  the  nature  of  the  operation  that  theore- 
tically, the  curve  would  never  touch  the  upper  line. 

Under  the  assumptions  of  constant  high-pressure  cut-off,  the  engine 
load  must  gradually  increase  as  the  receiver  pressure  is  built  up.  For 
constant  load,  the  high-pressure  cut-off  would  gradually  shorten,  length- 
ening the  time  required  to  regain  equilibrium,  as  less  steam  is  exhausted 
into  the  receiver  with  each  stroke. 

On  engines  governing  by  method  (2)  there  is  usually  a  hand  adjust- 
ment for  the  low-pressure  cut-off.  If  there  is  a  considerable  load  change, 
being  fairly  constant  for  quite  a  period  during  the  day,  the  cut-off  may 
then  be  adjusted  so  as  to  keep  a  more  uniform  receiver  pressure  and  a 
more  even  division  of  work  between  the  cylinders. 


43.3 


20.2 


4  6  8 

Number  of  Strokes 

FIG.  108. 


\Z 


14 


In  starting  a  compound  engine,  the  initial  pressure  in  the  receiver  is 
approximately  that  of  the  atmosphere.  The  pressure  is  gradually 
built  up  as  in  the  case  of  changing  load,  but  requires  a  longer  time.  To 
hasten  pressure  increase,  the  low-pressure  cut-off  may  be  shortened  by 
the  hand  adjustment,  and  gradually  lengthened  again  as  the  pressure 
rises,  giving  a  better  work  distribution  between  the  cylinders. 

In  engines  governing  by  method  (1)  there  is  usually  a  provision  for  ad- 
justing the  relative  high-pressure  and  low-pressure  cut-offs,  making  it 
possible  to  obtain  any  desired  work  division.  On  some  engines  there  is 
also  an  independent  adjustment  for  starting,  by  means  of  which  the 
low-pressure  gear  connections  may  be  temporarily  broken  and  a  short 
low-pressure  cut-off  obtained;  then  when  equilibrium  is  attained  the  con- 
nection is  again  made  and  both  cylinders  controlled  by  the  governor. 
These  various  appliances  are  illustrated  in  Chap.  XX, 


THE  COMPOUND  STEAM  ENGINE  175 

The  relative  merits  of  the  two  methods  depend  somewhat  upon  con- 
ditions. Both  are  used  on  high-grade  engines.  For  engines  with  large 
overload  capacity,  especially  when  the  overload  may  come  on  unexpect- 
edly and  remain  for  an  appreciable  period,  method  (1)  is  probably  pref- 
erable. For  comparatively  small  ranges  of  load,  it  is  simpler  to  govern 
only  the  high-pressure  cylinder.  With  a  large  receiver  and  a  sensitive 
governor  this  method  will  give  good  results  even  with  fluctuating  loads*. 

For  the  rated  power,  the  terminal  pressure  in  the  low-pressure  cylinder 
is  usually  selected,  presumably  so  as  to  give  maximum  economy.  This 
may  vary  from  2  to  4  Ib.  above  back  pressure,  but  should  not  be  below 
5  Ib.  absolute. 

69..  Maximum  Thrust.  —  In  Figs.  96  and  99  it  may  be  seen  that  the 
maximum  pressure  in  the  low-pressure  cylinder  is  greater  than  the  pres- 
sure at  cut-off,  which,  returning  to  the  notation  used  previous  to  Par.  68, 
is  denoted  by  P2.  Let  the  ratio  of  the  maximum  absolute  pressure  to 
PI  be  denoted  by  w,  then  the  maximum  piston  thrust  may  be  determined. 

Tandem  Compound.  The  thrust  of  both  high-pressure  and  low-pres- 
sure pistons  is  taken  by  the  main  piston  rod  and  transmitted  to  the 
other  engine  parts.  Let  this  be  denoted  by  Px'}  then,  neglecting  the 
effect  of  inertia  of  reciprocating  parts: 


Px  =          -i  -  rnPz)  +          (mP2  -  P.) 

(34) 


Cross-compound.  —  The  maximum  thrust  of  the  high-pressure   piston 
is,  from  Par.  66: 

Px  =  (Pi  -  P,)  ^  X35) 

The  influence  of  the  receiver  is  to  reduce  this  somewhat,  as  may  be  seen 
in  Figs.  99  and  102,  but  may  be  neglected  on  the  side  of  safety. 
The  maximum  low-pressure  thrust  is: 

Px  =          (wiP«  -  P3)  (36) 


The  influence  of  the  receiver  may  not  safely  be  neglected  and  is  provided 
for  by  the  factor  m,  to  be  derived  presently. 

The  parts  of  the  high-  and  low-pressure  engines  are  alike,  therefore,  the 
greater  value  given  by  Formulas  (35)  and  (36)  must  be  used.  Except  for 
small  values  of  R  or  when  very  large  receivers  are  employed,  the  maximum 
value  of  Px  is  given  by  (36),  and  should  this  be  much  in  excess,  the  high- 
pressure  piston  rod  may  be  determined  to  resist  the  thrust  given  by  (35). 


176 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


With  tandem  engines  with  the  low-pressure  cylinder  next  the  frame — 
the  more  usual  arrangement — (35)  may  be  used  for  the  high-pressure 
rod;  should  the  high-pressure  cylinder  be  next  the  frame,  the  low- 
pressure  rod  must  be  found  from  (36)  and  the  main  rod  from  (34). 

The  value  of  m  for  a  tandem  engine  may  be  found  by  taking  the  ratio 
P3/P4,  Fig.  95,  from  (s).  Substituting  the  notation  of  Par.  66  and  letting 
q  be  the  ratio  of  the  receiver  volume  to  volume  of  stroke  of  the  low-pres- 
sure cylinder: 


m 


(37) 


R 


It  is  obvious  from  (37)  that  m  increases  with  /2  but  not  in  the  same 
ratio;    the  maximum  low-pressure  cut-off  must  therefore  be  assumed. 

Table  26  is  computed  from  (37)  for  y2  cut-off,  assuming  that  ki  = 
k2  =  0.04,  and  with  values  of  q  ranging  from  0.25  to  1.5,  and  R  from  3 
to  8.     Table  27  is  for.  a        cut-off. 


TABLE    26 


0.25 

0.5 

0.75 

i 

1.5 

3 

1.53 

1.38 

.30 

1.24 

1.18 

4 

1.69 

.47 

.36 

1.29 

1.21 

5 

1.81 

.54 

.41 

1.32 

1.23 

6 

1.90 

.59 

.44 

1.35 

1.25 

7 

.1.98 

.63 

.46 

1.37 

1.26 

8 

2.05 

.67 

.48 

1.38 

1.27 

TABLE  27 


Q 

0.25 

0.5 

0.75 

i 

1.5 

3 

1.80 

1.57 

1.45 

.36 

.27 

4 

2.04 

1.71 

.54 

.44 

.32 

5 

2.22 

1.81 

.62 

.48 

.35 

6 

2.35 

1.89 

.66 

.53 

.38 

7 

2.47 

1.95 

.69 

.56 

.39 

8 

2.58 

1.99 

.72 

.57 

.41 

THE  COMPOUND  STEAM  ENGINE 


177 


For  a  cross-compound  engine  the  maximum  value  of  m  is  found  under 
Case  1,  by  the  ratio  of  (s)  to  (p),  which  gives,  when  I*  ^  0.5: 

1  +  kl  -  Xi 


..I 


R(q 


0.5 


+  1   \(q  +  A*  + 


+  0.5 
R 


(38) 


Comparing  Fig.  97  with  Fig.  103,  it  is  obvious  that  the  maximum 
value  of  m,  neglecting  angularity  of  the  connecting  rod,  occurs  when  lz 
=  0.5;  substituting  this  in  (38)  and  assuming  that  ki  =  kz=  0.04, 
and  that  Xi  =  0.8,  Table  28  may  be  computed  for  different  values  of  q 
and  R. 

TABLE  28 


Q 

R 

0.25 

0.5 

0.75 

1 

1.5 

3 

2.03 

1.63 

1.45 

1.36 

1.25 

4 

2.15 

1.69 

.49 

1.38 

1.27 

5 

2.23 

1.73 

.52 

1.40 

1.28 

6 

2.30 

1.76 

.54 

1.41 

1.29 

7 

2.35 

1.78 

.55 

1.42 

1.29 

8 

2.38 

1.80 

.56 

1.43 

1.30 

The  slight  rise  in  pressure  shown  in  the  cross-compound  low-pressure  dia- 
grams at  the  beginning  of  the  stroke  may  be  neglected.  This  rise  is 
due  to  the  fact  that  the  high-pressure  piston  is  near  mid-stroke,  traveling 
at  about  maximum  velocity  while  the  low-pressure  piston  is  leaving  the 
end  of  the  stroke;  the  high-pressure  piston  is  displacing  volume  more 
rapidly  for  a  short  time  than  the  low-pressure  piston,  causing  the  rise 
in  pressure,  usually  not  more  than  1  pound. 

The  effect  of  inertia  of  the  reciprocating  parts  has  the  same  effect 
as  with  the  simple  engine.  In  the  latter  it  was  neglected  when  maxi- 
mum thrust  was  considered  as  explained  in  Par.  59,  Chap.  XII.  In  the 
compound  it  tends  to  offset  the  effect  of  m,  but  except  in  high-speed 
engines  it  may  be  neglected,  which  is  usually  on  the  side  of  safety.  In 
any  engine,  the  inertia  of  the  piston  only  should  be  deducted  from  the 
effect  on  the  piston  rod,  the  inertia  of  piston  and  rod  from  the  effect  on 
the  crosshead  and  so  on.  In  view  of  this,  and  the  possibility  of  more 
or  less  erratic  action  of  a  compound  engine,  inertia  may  usually  be 
neglected  except  in  its  effect  upon  turning  effort. 

Formulas  (34)  to  (36)  may  be  used  in  determining  the  engine  parts 

12 


178  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

of  compound  engines,  selecting  ra  from  Tables  26,  27  or  28;  should  R 
lie  between  two  tabular  values  the  next  higher  may  be  taken. 

It  is  sometimes  said  that  a  tandem  compound  requires  no  receiver 
other  than  the  connecting  piping;  a  comparison  of  Tables  26  and  27  with 
28  will  show  that  if  extreme  maximum  thrust  is  to  be  avoided,  the  require- 
ments are  practically  the  same  as  for  a  cross-compound  engine,  especially 
for  long-range  cut-off. 

70.  Indicated  Horsepower. — If  the  power  is  to  be  determined  from  a 
test,  general  Formula  (17),  Chap.  XII  may  be  applied  to  each  cylinder, 
the  sum  being  the  total  power.  For  the  purpose  of  design,  (22)  to  (25), 
Chap.  XII  may  be  applied. 

Case  1. — Single-acting,  2-cylinder  compound. 

=  PHD^S      PLDL*S 
84,000  ""  84,000 

;       \  .    ':      -(£  +  '•)  ^ 

From  which: 

(40) 


From  (32): 

D, 


Case  2.  —  Double-acting,  2-cylinder  compound. 


DL  =  213       __   _  (43) 


From  which: 


DH  may  be  found  from  (41). 
From  (22),  Chap.  XII: 

-  ;;  -       ;'  i  *=™    '    ,:  *    '  v;,.  (44) 

where  L  is  the  stroke  in  inches. 

If  PH  and  PL  are  from  theoretical  diagrams  they  should  be  multiplied 
by  a  diagram  factor.  In  Par.  57,  Chap.  XII,  a  quotation  from  Kent's 
Mechanical  Engineers'  Pocket  Book  is  given,  with  a  portion  of  a  table 
of  diagram  factors  from  Seaton.  The  remainder  of  the  table,  applying 
to  marine  engines  is  as  follows:  * 


THE  COMPOUND  STEAM  ENGINE  179 

Particulars  of  engine  Factor 

Compound  engines  with  expansion  valve  to  h.-p.  cylinder,  jacketed, 
with  large  ports,  etc.  0  .  9  to  0  .  92 

Compound  engines  with  ordinary  slide  valves,  -cylinders  jacketed, 
good  ports,  etc  ...................................................   0.8  to  0.85 

Compound  engines  as  in  early  practice  in  the  merchant  service,  with- 
out jackets  and  expansion  valves  .............  .....................   0.7  to  0.8 

Fast  running  engines  of  the  type  and  design  usually  fitted  in  war  ships  0.6  to  0  .  8. 

Creighton  gives  the  following  diagram  factors  for  stationary  com- 
pound engines: 

Kind  of  engine  Factor 

High  speed,  short  stroke,  unjacketed  ...........................  0.60  to  0.80 

Slow  rotative  speed,  unjacketed  ................................  0.70  to  0.85 

Slow  rotative  speed,  jacketed  ..................................  0.85  to  0.90 

Corliss  .......................................  ...............  0.85  to  0.9Q 

Triple-expansion  ..........................  .  ..................  0.60  to  0.70 

Bauer  and  Robertson  give  for  marine  compounds: 

Kind  of  engine  Factor 

Large  engines  up  to  100  r.p.m  ..........................  .  .......  0.  60  to  0.  67 

Small  engines  greater  than  100  r.p.m  ...........................  0.55  to  0.60 

Triple-expansion  —  war  vessels  with  high  r.p.m  ..................  0.53  to  0.54 

Triple-expansion  —  merchant  vessels  up  to  100  r.p.m  .............  0.56  to  0.61 

71.  Theoretical  steam  consumption  may  be  calculated  as  in  Par.  61, 
Chap.  XII.  The  weight  of  steam  is  measured  by  the  high-pressure  cylin- 
der; then,  referring  the  low-pressure  m.e.p.  to  the  high-pressure  cylinder, 
(26),  Chap.  XII  becomes: 


W  =  p+pL  MIi  +  fc)  -  «fc(l  +  *i  -  *0]  (45) 

where  W  is  the  water  rate  in  pounds  per  horsepower-hour,  w,  the  weight 
per  cubic  foot  of  steam  at  initial  pressure  and  WB  its  weight  at  back 
pressure. 

72.  Standard  Engines.  —  Standard  engine  parts  may  be  selected  for 
compound  engines  in  the  same  manner  as  for  simple  engines  working 
under  different  pressures,  as  explained  in  Par.  63,  Chap.  XII,  by  equating 
tiie  maximum  thrust  of  the  standard  engine  with  that  of  the  compound. 

For  the  tandem  compound: 


From  which: 


MP 


__  -=KDL  (46) 


180  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

For  the  cross-compound : 

High-pressure  cylinder  \P^  -  Ir2 

from  (35),  D*  =  D*  \~~~     =  KL>L 


Low-pressure  cylinder               ~         ~      /mP2  —  P3        T^^ 
from  (36),  D*  =  D*y ~p—       =  KD^ 

As  explained  in  Par.  69,  the  greater  value  must  be  used. 

Values  of  K  may  be  tabulated  for  given  pressure  ranges,  receiver 
ratios,  etc.,  and  it  will  be  found  that  cylinder  ratio  has  little  effect  on  power 
if  within  practical  limits. 

73.  Application  of  the  formulas  of  this  chapter  to  design  will  be 
shown  by  a  number  of  examples.  As  noncondensing  compounds  may 
be  designed  by  the  simple  method  of  Par.  65,  the  examples  will  be  confined 
tto  the  more  important  condensing  compound. 

Example  1.— Let  PI  =  165,  P3  =  2,  PK  =  4,  p2  =  6,  #1  =  0.9,  z2  =  0.8 
and  fci  =  k2  =  0.04. 
From  (16): 

/165 


\   4 

From  (17): 

RK2  =  82.5. 

From  (18): 

P    ~            165                "041b 

JT  2    —       ,  .        ~—           rr          —   jiU.rr   ID. 

\/6.42  X  82.5 

From  (19): 

,0.14       2X0.24_ 

"  6.42           20.4 

From  (20)  : 

20.4 

r2  =  —  ^-  =  3.4. 

From  (21): 

1  04 

12  =  —^  -  0.04  =  0.266. 

From  (22)  : 

20.4  X  6.42  X  0.304 

165 

From  (23)  : 

1.04 

"  0.24  ~ 

From  (24)  : 

0.201. 


165       0_  _ 
rT  =  —  =  27.5 


THE  COMPOUND  STEAM  ENGINE  181 

From  (27) : 

°-14  -  3.5. 


From  (28) : 

_0.24 
Tcz  ~  0.04  = 
From  (29) : 

PH  «  165[(0.24  X  2.458)  -  0.04]  -  20.4[0.9  + 

(0.04  X  3.5  X  1.2528)] 
=  69.4  Ib. 
From  (30): 

PL  =  20.4[(0.306  X  2.224)  -  0.04]  -  2[0.8  + 

(0.04  X  6  X  1.791)] 
=  10.6  Ib. 
The  high-pressure  terminal  drop  given  by  (31)  is: 

di  =  ^|  -  20.4  =  18  Ib. 
4.3 

The  total  m.e.p.  referred  to  the  low-pressure  cylinder  is: 


If  H  =  1000,  S  =  800  and  the  diagram  factor  is  0.85,  (43)  gives: 
dL  =  56",    and  from   (41),  Dn  =  22".     If  L  be  taken  as  48",  N  =  100. 
The  size  of  the  engine  is  then: 

22"  and  56"  by  48"  -  100. 

The  ratio  of  low-pressure  to  high-pressure  work  is  given  by    (33), 
and  is: 

• 


The  cylinder  ratio  in  Example  1  is  above  average  practice.  High 
ratios  have  been  objected  to  as  lacking  overload  capacity.  Assuming  a 
cut-off  of  Y±  hi  each  cylinder,  P2,  found  from  (26),  is  24.54.  Substituting 
these  values  in  (29)  and  (30),  gives:  PH  =  133.6  and  PL  =  21.34.  The 
ratio  of  low-  to  high-pressure  work  is: 

HL  _  21.34  X  6.42  _ 
H~H=          133.6 

The  m.e.p.  referred  to  the  low-pressure  cylinder  is: 

+  21.34  =  42.14 


182  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Then: 


and  the  overload  capacity  is  97  per  cent.,  which  seems  ample. 

Assuming  a  receiver  volume  equal  to  0.5  the  volume  of  stroke  of  the 
low-pressure  cylinder,  the  maximum  value  of  m  from  Table  28  is  1.78 
for  a  cross-compound  engine.  Then  the  standard  engine  having  the 
same  parts  as  the  compound,  has  from  (47)  for  the  high-pressure  side, 
a  diameter  of: 


Ds  =  ft,..pg~  =  1.0751)* 
if  pS}  the  standard  pressure  is  125.     As 

D"  =  ^A2  Ds  =  °'423Di- 
For  the  low-pressure  side,  from  (48) : 


This  is  larger  and  should  be  used.  A  small  receiver  was  taken  for  safety, 
as  such  a  receiver  might  be  used;  but  for  special  cases,  if  standard  com- 
pound engine  tables  are  not  to  be  used,  actual  values  of  q  may  be  taken. 

D    -  '''" 


This  value  is  still  larger  than  for  the  high-pressure  cylinder  and  should 
be  used  in  this  case  if  q  is  no  smaller  than  1. 

For  the  tandem  compound  with  a  maximum  cut-off  of  %  stroke  and 
q  =  0.5,  m  =  1.95;  and  from  (46): 


-  =  0.6770. 


If  q  =  1,     Ds  =  0.636DL. 

Example  2.  —  Assume  the  same  data  as  before  except  PK,  which  is 
3  —  the  minimum  extreme  given.  Without  taking  step  by  step  as  was 
done  in  Example  1,  the  principal  results  are: 

R  =  7.42,  P2  =  19.45,  12  =  0.281,  Zi  =  0.237,  PH  =  77.7,  PL  =  10.34, 
Also: 

~  =  0.988  ~  +  H  =  20.82  d*  =  24.55. 

HH  K> 

For  q  =  0.5,  D^  =  0.513DL  for  cross-compound,  and  0.656DL  for  tandem. 


THE  COMPOUND  STEAM  ENGINE 


183 


Example  3.  —  The  same  as  1  and  2  except  PK  is  6,  the  maximum  ex- 

treme given. 

Then: 

;     R  =  5.25,  Pz  =  21.8,  12  =  0.246,  h  =  0.161,  Pa  =  57.6,  PL  = 

10.82. 

Also: 

77                                                                                           Prr 

^  =  0.986.                V  +  ^  =  21-77-                dl  =  10-1- 
/i^                                      /t 

For  q  =  0.5,  Ds  =  0.54DL  for  cross-compound,  and  0.72Z)L  for  tandem. 
If  q  is  1.5  for  this  cylinder  ratio,  the  high-pressure  cylinder  gives  a  greater 
thrust  than  the  low-pressure;  it  is  safe  to  try  both  (47)  and  (48)  for 
cross-compound  engines. 

It  may  be  seen  that  throughout  the  entire  range,  Formula  (18)  gives 
very  close  results,  and  that  the  maximum  variation  of  the  total  m.e.p. 
referred  to  the  low-pressure  cylinder  is  0.95  lb.,  or  4.5  per  cent,  of  the 
smallest  value.  Due  to  reasons  explained  in  Par.  47,  Chap.  IX,  the  actual 
variation  would  probably  be  even  less;  this  shows  that  the  power  of  a 
compound  engine  depends  mainly  upon  the  low-pressure  cylinder,  the 
cylinder  ratio  affecting  it  but  little.  The  m.e.p.'s  of  the  examples  given 
must  be  multiplied  by  a  diagram  factor  before  using  in  the  power  formula. 

Taking  DL  =  56  as  just  calculated,  q  =  0.5,  and  the  nearest  safe  whole 
number  for  Ds,  Table  29  is  given  so  that  comparison  may  be  easily  made. 

TABLE  29 


Ds 

Example 

PK 

R 

DH 

DL 

C.C. 

Tandem 

2 

3 

7.42 

20 

56 

29 

37 

1 

4 

6.42 

22 

56 

30 

38 

3 

6 

5.25 

24 

56 

31 

41 

Ds  increases  as  the  value  of  R  decreases,  and  this'  is  a  measure  of  the 
weight  and  cost  of  the  engine;  it  also  probably  has  some  influence  on  the 
friction.  A  slightly  greater  maximum  power  may  be  obtained  from  the 
lower  ratio  when  the  low-pressure  is  of  the  same  diameter  in  all  cases, 
but  ample  overload  may  be  carried  by  engines  of  higher  ratio  as  shown  in 
Example  1. 

Example  4. — A  comparison  will  now  be  made  with  a  Nordberg  engine 
referred  to  in  their  Bulletin.  It  is  a  19  and  44  by  42  in.  cross-compound 
engine,  and  the  cylinder  ratio  from  (32)  is: 


184  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

For  the  particular  test  mentioned:  Pi  =  169.18,  P3  =  1.91  and  the 
receiver  pressure,  which  would  be  somewhere  between  P2  and  raP2, 
was  20.75  Ib.  Actual  m.e.p.  's  were 

fiPH  =  52.605,  and/2PL  =  9.557. 

The  total  m.e.p.  referred  to  the  low-pressure  cylinder  was  19.352. 

Clearance  and  compression  are  not  given,  and  it  will  be  assumed  that 

ki  =  0.07,  &2  =  0.05,  xi  =  0.9,  and  z2  =  0.8;  also  that  p2  =  6  Ib. 

With  the  above  data,  the  results  from  the  formulas  of  Par.  66  are: 
P2  =  21.7,  Z2  =  0.24,  h  =  0.14,  PH  =  61.4  and  PL  =  11.21. 

Using  a  diagram  factor  of  0.85  : 

fiPH  =  52.2  and  /2PL  =  9.55. 

The  total  m.e.p.  referred  to  the  low-pressure  cylinder  is  19.28,  differing 
from  the  test  value  by  less  than  0.5  per  cent. 

Assuming  a  receiver  volume  1.5  times  the  volume  of  stroke  of  the  low- 
pressure  cylinder,  (38)  gives:  m  =  1.028.  The  maximum  receiver  pres- 
sure is  then  : 

1.028  X  21.7  =  22.3. 


This  is  1.55  Ib.  greater  than  the  actual  receiver  pressure  given,  a  result 
to  be  expected  due  to  shrinkage  of  the  diagram  as  shown  by  the  diagram 
factor. 

The  data  of  Example  1  is  plotted  in  Figs.  96  and  99,  the  dotted  lines 
for  an  indefinitely  large  receiver  and  the  full  lines  showing  the  effect  of  a 
receiver  of  practical  volume. 


CHAPTER  XIV 
THE  INTERNAL-COMBUSTION  ENGINE 

74.  Introduction. — In  the  treatment  of  this  chapter  a  knowledge  of  the 
contents  of  Chaps.  V  and  VI  is  assumed.  A  thorough  understanding  of 
the  principles  of  operation  of  the  internal-combustion  engine  can  not  be  had 
without  following  through  the  power  formulas  based  upon  heat  quantities; 
but  certain  practical  formulas  have  value  and  are  derived  in  this  chapter; 
their  constants  are  also  given  in  terms  of  the  quantities  in  the  more 
theoretical  formulas. 

Notation. 

h  =  heating  value  (high  or  low)  in  B.t.u.  per  pound  of  liquid  fuel, 
or  per  cubic  foot  of  gaseous  fuel  at  standard  temperature  and 
pressure. 

a  =  actual  air  supply  in  cubic  feet  per  pound  of  liquid  fuel,  or 
per  cubic  foot  of  gaseous  fuel,  at  standard  temperature  and 
pressure. 
eM  =  mechanical  efficiency  of  engine. 

6  =  indicated  thermal  efficiency  of  engine. 
eB  =  thermal  efficiency  referred   to   brake   horsepower;  called   by 

Gtildner,  economic  efficiency.  • 

ev  =  volumetric  efficiency — the  ratio  of  the  volume  of  the  charge 
(fresh  air  and  fuel)  at  standard  temperature  and  pressure,  to 
the  volume  of  stroke.     See  Chap.  VI,  Par.  27. 
PM  =  mean  effective  pressure  in  pounds  per  square  inch. 
p  —  unbalanced  pressure  in  pounds  per  square  inch  at  any  point 

of  stroke. 

px  =  the  maximum  value  of  p. 

P  =  total  unbalanced  pressure  at  any  point  of  stroke. 
Px  =  maximum  value  of  P. 
D  =  diameter  of  cylinder  bore  in  inches. 
L  =  length  of  piston  stroke  in  inches. 
N  =  r.p.m. 

S  =  mean  piston  speed  in  feet  per  minute, 
ra  =  number  of  strokes  per  cycle. 
H  —  horsepower  in  general. 

185 


186  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

HI  =  indicated  horsepower  per  working  cylinder  end. 
HB  '-=  brake  horsepower  per  working  cylinder  end. 
HM  =  maximum  horsepower,  indicated  or  brake. 
HR  =  rated  horsepower,  indicated  or  brake. 


=  JT" 
HR 

=  liquid  fuel  used  in  pounds  per  hour. 

=  gaseous  fuel  used  in  pounds  per  hour,  at  standard  temperature 
and    ressure. 


WF 
VF 

and  pressure. 

75.  Mean  Effective  Pressure.  —  The  absence  of  accurately  predeter- 
mined pressures  possible  in  the  steam  engine  makes  the  study  of  indicator 
diagrams  for  the  internal-combustion  engine  more  difficult,  and  a  good 
collection   of  diagrams  more  desirable.     These  diagrams  are  useful  for 
determining  the  proper  yalve  setting,  timing  of  ignition,  best  fuel  mixture, 
and  for  finding  the  output  and  efficiency  of  the  engine.     However,  the 
uncertainty  of  limiting  pressures,  the  lack  of  uniformity  of  assumptions 
for  a  theoretical  diagram  and  the  wide  deviation  of  the  actual  from  such 
a  reference  diagram,  makes  the  use  of  the  diagram  factor  (see  Chap.  XII, 
Par.  57)  for  determining  the  m.e.p.  less  satisfactory  than  for  the  steam 
engine  and  other  methods  of  calculation  are  more  in  favor. 

In  Par.  29,  Chap.  VI,  the  following  formulas  are  derived  from  a  more 
general  discussion  in  Par.  23  of  the  same  chapter. 

For  gaseous  fuel  PM  =  5Aeev      ,    .  (1) 

For  liquid  fuel.  PM  =  5Aeev  -  (2) 

These  apply  to  both  2-stroke  and  4-stroke  cycles,  the  difference  in 
the  values  of  PM  for  the  two  cycles  being  usually  due  to  the  different 
values  of  e  and  ev,  the  indicated  thermal  efficiency  and  volumetric  effi- 
ciency respectively. 

As  explained  in  Chap.  VI,  this  is  the  mean  pressure  which,  if  exerted 
throughout  one  stroke  of  the  piston,  would  do  the  work  of  the  cycle  for 
one  working  cylinder  end. 

76.  Horsepower.  —  In  Chap.  VI,  Par.  23,  the  following  general  formula 
for  indicated  horsepower  is  derived  for  one  cylinder  end: 

„  _  2  X  144    PMvaN  ,ox 

:    33,000       ~^T 

where  N  is  the  r.p.m.,  m  the  number  of  strokes  per  cycle,  and  vs  the 
volume  of  stroke  in  cubic  feet. 


THE  INTERNAL-COMBUSTION  ENGINE  187 

If  L  is  the  length  of  stroke  in  inches,  and  D  the  diameter  of  the  cylin- 
der bore  in  inches: 

7T       D2        L 

Vs  ~  4  *  144  "  12 
Substituting  in  (3)  gives: 

PMD*LN        PMD*S 
1      252,100m      42,000m 

where  S  is  the  mean  piston  speed  in  feet  per  minute,  which  is: 

7      P^-f'ISS     (5) 

The  brake  horsepower  is: 


„  „       eMPMD2LN         MM  ... 

euHl  ~  252,100m  ==  42,000m 

For  gaseous  fuel,  substituting  (1)  in  (4)  and  (6)  gives: 

h         D2LN  h         D2S 

'F^~TT  '  46,700  m  =  e€v  a  +  1  '  7790  m 
And  as  eAfe  =  eB: 

„  ft         D*LN  h         D2S  ,_. 

#*  =  ^^^'46j6^  =  ^r^n-779^ 

From  (7)  and  (8)  : 


D  =  88.25  JmHl(a+l)  =  88.25  >g^a  +  D  (9) 

\       eeF£/i  \      eBevSh 

For  liquid  fuel,  substituting  (2)  in  (4)  and  (6)  gives: 
Ri  =  eeyh.    D2LN    =  eevh      D2S 

h     D*LN  h      D2S 

HB=eBeva'WlW^  =  €Beva'^^  (1]) 


From  (10)  and  (11): 

Z)=  88.25^g|  =  88.25 

Although  derived  in  a  different  manner,  these  equations  give  results 
identical  with  the  formulas  of  Giildner,  which  are  in  terms  of  brake 
horsepower. 

Formulas  (9)  and  (12),  being  used  for  design,  are  in  terms  of  piston 
speed  rather  than  stroke  and  revolutions.  Some  discussion  of  piston 
speed  is  given  in  Par.  48,  Chap.  IX;  with  proper  design  there  seems  to  be  no 
restrictions  within  reasonable  limits,  other  than  those  of  gas  velocity, 
which  will  be  discussed  in  Chap.  XX. 


188 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  relation  of  S,  L  and  N  may  be  adjusted  by  means  of  (5)  when 
the  cylinder  diameter  has  been  determined;  if  N  is  fixed  by  reason  of 
service  considerations,  L  may  be  found.  If  not,  the  question  is  one  of 
the  ratio  of  stroke  to  diameter;  this  ranges  from  1  to  2,  Diesel  engines 
commonly  having  the  ratio  1.5.  Some  discussion  of  this  ratio  is  given 
in  Par.  47,  Chap.  IX,  under  Design  of  Cylinders,  and  insofar  as  it  concerns 
the  weight  of  the  engine  parts  it  applies  to  internal-combustion  engines. 
For  an  extensive  discussion  of  the  subject,  the  reader  is  referred  to  Giild- 
ner's  Internal-combustion  Engines. 

The  heating  value  of  fuel  and  theoretical  air  supply  are  given  in  Table 
30.  Also  see  Par.  42,  Chap.  VIII.  The  actual  air  supply  must  be  greater 
than  the  theoretical  to  insure  perfect  combustion;  aside  from  this  there 
should  be  an  excess  of  air  to  provide  for  an  increase  of  fuel  supply  for 
overloads.  Even  if  an  engine  is  rated  at  its  maximum  capacity  there  are 
enough  uncertain  factors  in  its  design  to  make  it  unsafe  to  allow  too  small 
a  margin  over  the  air  theoretically  required  for  perfect  combustion. 

Table  30  is  abridged  from  Guldner's  Internal-combustion  Engines,  and 

TABLE  30 


Fuel 

h  (low)  per 

a  (theoreti- 
cal) per 

a  (actual) 
per 

eB  for  HB  per  cylinder  end 
of 

Cu. 

ft. 

Lb. 

Cu. 
ft. 

Lb. 

Cu. 

ft. 

Lb. 

5 

10 

25 

50 

100 
and 
over 

Illuminating  gas  —  lean  

505 

5.5 

7.5 

0.20 

0.22 

0.24 

0.26 

0.27 

Illuminating   gas  —  rich  

675 

6.5 

10.0 

0.20 

0.22 

0.24 

0.26 

0.27 

1.1 

Producer    gas  —  anthracite.  .  .  . 

141 

0.85 

to 

0.17 

0.19 

0.21 

0.23 

0.24 

1.4 

1.0 

Blast-furnace  gas  

106 

0.75 

to 

0.18 

0.20 

0.22 

0.24 

1.2 

Coke-oven  gas         

505 

5.3 

7.0 

. 

0.17 

0.19 

0.21 

0.23 

257 

Kerosene  

18,900 

185.0 

to 

0.11 

0.12 

0.13 

353 

288 

Crude  oil  —  Diesel 

18,000 

176.0 

to 

0.25 

0.26 

0.27 

0.30 

0.315 

323 

240 

Gasoline                         

18,500 

176.0 

to 

0.19 

0.21 

0.23 

323 

128 

Alcohol  (90  per  cent,  vol.)  

10,300 

96.5 

to 

0.22 

0.24 

0.26 

193 

may  be  used  as  a  guide  in  the  selection  of  h,  a  and  eB.     Values  of  HR  are 
for  one  cylinder  end.     Both  theoretical  and  practical  values  of  air  supply 


THE  INTERNAL-COMBUSTION  ENGINE  189 

are  given,  the  latter  being  based  upon  an  overload  capacity  of  from  15 
to  20  per  cent. 

Volumetric  efficiency  is  discussed  in  Par.  27,  Chap.  VI;  some  practical 
values  from  Gtildner  are  given  in  Table  31.     The  values  from  Guldner's 

TABLE  31 

Type  of  engine  ev 


Slow-speed,  mechanically  operated  inlet  valve . . 

Slow-speed,  automatic  inlet  valve 

High-speed,  mechanically  operated  inlet  valve. . 
High-speed,  automatic  inlet  valve 


0.88  to  0.93 
0.80  to  0.87 
0.78  to  0.85 
0.65  to  0.75 


tables  are  based  upon  a  standard  pressure  of  14.7  Ib.  per  square  inch  and 
32  degrees  F.,  although  he  recommends  28.92  in.  of  mercury  and  59  degrees 
F.  The  A.S.M.E.  Code  adopts  30  in.  of  mercury  and  60  degrees  F. 

The  mechanical  efficiency  eM  varies  from  0.6  to  0.9  and  is  discussed 
in  Chap.  X. 

Simple  Formulas.  —  Thus  far  the  formulas  have  been  of  a  general 
character  and  these  are  necessary  for  a  comprehensive  study  of  engine 
power;  but  by  making  substitution  of  experimental  values  of  PM  for 
rated  power  in  (6),  or  of  h,  a,  ev  and  eB  in  (8)  and  (11),  a  simple  formula 
may  be  derived.  E.  W.  Roberts,  in  The  Gas  Engine  Handbook  gives 
(the  notation  being  changed)  : 


.,„      ., 

in  which  the  constant  K  differs  for  different  fuel  and  with  the  type  of  en- 
gine.    This  may  be  written: 


Then: 

D  =  0.41^^i  (15) 

Comparing  (14)  with  (6)  and  (8),  the  value  of  K  may  be  expressed 
in  terms  of  the  quantities  in  these  formulas;  or: 

K  =  MI^  (16) 

also: 

K  =  46>7QQm(a  +  1)  /17) 

eBevh 

As  a  -f  1  is  but  slightly  different  from  a  for  liquid  fuels,  (17)  may  be 
applied  to  all  internal-combustion  engines. 


190  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Roberts  gives  for  the  average  size  of  4-cycle  engine: 

For  natural  gas K  =  16,000; 

For  gasoline K  =  14,000; 

stating  that  K  becomes  smaller  as  the  cylinder  size  increases.     He  cites  a 
case  of  an  automobile  engine  with  exceptionally  large  valves  and  ball 
bearings  in  which  K  was  11,500.     Smaller  values  have  been  obtained — 
notably  in  airplane  engines  (see  Table  32). 
For  2-cycle  engines  Roberts  gives: 

For  gas K  =  12,000; 

For  gasoline K  =  10,000; 

For  large  blowing  engines K  =    8,400. 

It  is  claimed  that  these  values  are  approximate. 

The  Bruce-Macbeth  Engine  Co.  published  a  formula  for  4-cycle 
engines  which  gives : 

v       1,000,000 

K-  —p*- 

in  which  PB  varies  as  follows: 

For  natural  gas,  PB  =  75  lb.,  and  K  =  13,330; 
For  producer  gas,  PB  =  60  lb.,  and  K  =  16,660; 
For  gasoline,  PB  =  60  lb.,  and  K  =  16,660. 

A  comparison  of  this  formula  with  (16)  shows  that  PB  is  practically 
equal  to  eMPM,  the  brake  m.e.p.  The  indicated  m.e.p.  is  then  equal  to 
PB  divided  by  eM> 

Comparing  the  constants  of  4-cycle  and  2-cycle  engines  given  by 
Roberts,  the  2-cycle  has  but  40  per  cent,  greater  capacity  than  the  4- 
cycle.  Substituting  the  respective  values*  of  K  in  (17),  assuming  a  and 
h  the  same  for  both  engines,  the  value  of  the  product  of  eBev  for  the  4- 
cycle  engine  is  43  per  cent,  greater  than  for  the  2-cycle.  This  difference 
is  probably  in  part  due  to  eB,  but  largely  to  ev.  Comparing  the  value  of 
K  for  the  large  2-cycle  blowing  engines  mentioned  by  Roberts  with 
that  of  the  Bruce-Macbeth  producer-gas  engine  gives  practically  equal 
values  of  the  product  e^v-  This  is  no  doubt  due  to  the  greater  value 
of  ev  made  possible  by  the  separate  charging  cylinders  of  the  large  2-cycle 
engine;  this  is  probably  greater  than  for  the  4-cycle  engine,  while  eB 
is  smaller. 

Taking  K  from  engine  ratings  or  tests,  and  assuming  eB,  ev  and  h 
(or  taking  them  from  Table  30  or  from  tests),  the  air  supply  may  be 
determined  from  (17) ;  or: 

KeBevh        1 
0  "  46/700^  -  X 


THE  INTERNAL-COMBUSTION  ENGINE 


191 


A  comparison  of  engine  ratings  for  a  given  fuel  may  thus  be  made; 
the  smaller  the  air  excess  the  nearer  to  the  maximum  power  is  the  engine 
rated,  remembering  that  there  must  be  some  excess  air  at  maximum 
power.  Also  by  assuming  eM,  the  value  of  PM  may  be  found  from  (16). 

The  influence  of  ev  upon  capacity,  and  undoubtedly  upon  efficiency, 
is  considerable,  and  every  practical  means  available  for  increasing  this 
should  be  employed.  Ample  valve  openings,  easy  passages  and  proper 
timing  are  probably  the  chief  factors  in  a  high  value  of  ev.  From  Par.  27, 
Chap.  VI,  it  is  obvious  that  the  temperature  of  the  charge  directly  affects 
capacity  by  determining  in  part  the  value  of  ev;  the  other  factor  is  the 
pressure  of  the  charge,  which  depends  upon  valves,  passages,  etc.  as  just 
stated. 

The  value  of  eB  may  be  kept  a  maximum  by  proper  compression, 
carburetion,  ignition  and  lubrication,  in  addition  to  the  factors  just 
mentioned. 

Table  32  contains  some  data  pertaining  to  engines  of  different  type 
and  capacity,  taken  from  builders  catalogues,  published  results  of  tests, 
and  information  furnished  the  author  directly.  These  may  be  used  for 
comparison. 

TABLE  32 


Type 

m 

Number 
of 
cyl. 

HB  per 
cyl. 

D 
(in.) 

L 
(in.) 

L/D 

N 

s 

K 

CM 

PM 

Stationary  
Stationary  
Stationary  

4 
4 
4 

1 
1 
1 

1.5 
5.00 
7.00 

3M 
5M 
6 

4 
7 
8K 

.14 
.27 
.41 

550 
425 
375 

366 
496 
532 

18,000 
18,000 
16,400 

0.80 
0.80 
0.80 

70 
70 

78 

Stationary 

4 

1 

15.00 

8% 

14 

.62 

250 

584 

16,400 

0  80 

78 

Automobile 

4 

4 

10.00 

3H 

5K 

.61 

2,800 

2,450 

15,500 

0  80 

81 

• 

Automobile     .      .    . 

4 

6 

5.00 

3% 

4 

.10 

1,500 

1,000 

15,750 

0  78 

83 

Automobile  
Automobile  

4 
4 

6 
8 

7.50 
8.75 

3% 
3% 

4M 

5M 

.38 
.64 

2,600 
2,400 

1,950 
2,050 

17,500 
13,700 

0.78 
0  80 

74 
92 

Boat       

? 

1 

7.00 

*H 

4 

.84 

750 

500 

9,700 

0  74 

70 

Boat 

? 

1 

12.00 

5H 

5 

.87 

675 

563 

9,300 

0  76 

71 

Boat  
Boat  

4 
4 

4 
6 

7.50 
8.34 

4^ 
3% 

5H 
5K 

.27 
.40 

1,000 
1,300 

875 
1,140 

11,900 
11,550 

0.80 
0.78 

106 
112 

'3 
1 

Airplane  
Hot  bulb  
Hot  bulb  
Diesel 

4 
4 
4 
4 

6 
1 

1 
3 

20.00 
100.00 
140.00 
150  00 

4^ 
17 
20 
21 

6K 
27K 
34  H 
30 

.37 
.62 
.72 
.43 

1,350 
200 
165 
180 

1,465 
917 
950 
900 

9,900 
15,900 
16,250 
15,900 

0.78 
0.85 
0.85 
0  75 

130 
75 
73 

91 

Diesel 

4 

4 

125.00 

18% 

28% 

.50 

164 

775 

13,200 

0  73 

105 

Diesel 

? 

4 

200.00 

20 

36 

.80 

95 

570 

6,840 

0  70 

105 

O 

Natural  gas  
Natural  gas  (D.A.) 
Blast-furnace       gas 
(D.A.)  

4 
4 

4 

4 
2 

2 

37.50 
112.00 

500.00 

12K 
20 

45 

14 
36 

60 

.14 
.80 

1.33 

275 
125 

70 

640 
750 

700 

15,400 
16,100 

19,000 

0.80 
0.81 

0.84 

82 
78 

63 

Values  of  K  and  PM  have  been  calculated,  the  latter  by  assuming  a 
mechanical  efficiency  determined  by  Formulas  (10)  and  (11),  Chap.  X. 


192 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Until  recently  but  little  information  was  available  concerning  the 
Diesel  engine.  Much  valuable  information  is  contained  in  publications 
of  the  A.S.M.E.,  a  few  notes  from  the  paper  mentioned  at  the  end  of 
Chap.  V  being  given  at  this  place : 

Comparing  the  4-cycle  and  2-cycle  Diesel,  the  latter  gives  from  70 
to  80  per  cent,  more  power  for  the  same  cylinder  dimensions  and  speed. 
The  4-cycle  has  about  10  per  cent,  better  efficiency  and  the  mean  tem- 
perature is  lower.  In  this  country,  the  4-cycle  engine  is  built  in  sizes 
up  to  from  700  to  1000  b.  h.p.,  and  above  that  the  2-cycle  is  used. 

The  Diesel  engines  built  in  the  United  States  run  from  150  to  300 
r.p.m.,  with  piston  speeds  from  600  to  900  ft.  per  minute.  Some  high- 
speed marine  engines  run  as  high  as  480  r.p.m.  In  Europe  the  highest 
commercial  speed  is  from  350  to  400  r.p.m.,  with  as  high  as  550  for  sub- 
marines. 

Practically  all  Diesel  engines  are  single-acting,  and  nearly  all  in  this 
country  have  trunk  pistons  without  crossheads. 

The  mechanical  efficiency  at  full  load  is  about  75  per  cent,  for  4-cycle 
and  70  per  cent,  for  2-cycle  engines, 
i       Some  Diesel  engine  data  are  given  in  Table  32. 

The  m.e.p.  of  internal-combustion  engines  of  small  size  and  high  speed 
is  seldom  known;  it  is  sometimes  calculated  by  assuming  the  mechanical 
efficiency.  Some  values  of  PMj  also  of  h  are  given  in  connection  with  com- 
pression pressures  in  Table  35,  Par.  77,  which  may  be  compared  with  those 
already  given  in  Table  32.  E.  W.  Roberts,  in  The  Gas  Engine  of  October, 
1917,  gives  values  of  eMPM  for  a  number  of  engine  types,  which  represent 
recent  practice  in  the  United  States.  These  values  are  given  in  Table  33; 
values  of  eM  are  assumed,  from  which  the  values  of  PM  and  K  are  com- 
puted and  given  in  the  table. 

TABLE    33 


Type 

*MPM 

eM 

PM 

K 

Hot  bulb   2-cycle 

33 

0.70 

47 

15,300 

Producer  gas  

57 

0.80 

71 

17,700 

Gasoline   2-cycle                                     

60 

0.75 

80 

8,400 

Modern  high-speed  gasoline 

80 

0.80 

100 

12,600 

Modern  airplane  
Diesel.  4-cycle                                          

105 
75 

0.80 
0.73 

130 
103 

9,600 
13,500 

It  is  to  be  assumed  that  the  engines  are  4-cycle  unless  otherwise  stated. 
The  mechanical  efficiency  of  the  airplane  engine  is  perhaps  too  small; 
indications  are  that  a  much  better  value  has  been  obtained. 


THE  INTERNAL-COMBUSTION  ENGINE  193 

From  the  various  data  given  it  is  obvious  that  while  there  is  a  general 
agreement,  there  is  no  definite  standard  of  engine  rating,  and  if  there 
were  it  is  not  likely  that  capacities  found  from  tests  would  always  agree 
with  the  assumptions. 

With  new  work  it  is  well  to  determine  D  from  (9)  or  (12),  using  data 
from  Tables  30  and  31.  This  result  may  be  checked  by  (15),  taking  K 
from  Table  32  or  33;  or  by  (6),  taking  PM  from  Table  32,  33  or  35,  and  eM 
from  (10)  and  (11),  Chap.  X,  or  from  Table  32  or  33. 

With  assumed  values  of  eB,  ev  and  h,  the  theoretical  maximum  capacity 
limit;  is  when  a  is  the  theoretical  air  supply  required  for  perfect  combustion. 
The  actual  maximum  power  will  be  obtained  with  a  value  of  a  greater 
than  this,  the  power  always  being  less  than  the  theoretical  maximum. 
It  is  apparent  that  the  more  perfect  the  mixture  and  ignition,  the  smaller 
will  be  the  excess  of  air  required. 

Should  Formula  (6)  or  (15)  be  used  for  determining  cylinder  dimen- 
sions, the  air  supply  should  be  checked  by  (18),  and  if  equal  to,  or  less 
than  the  theoretical  minimum,  the  engine  will  not  develop  the  power 
desired — assuming  that  eB}  ev  and  h  have  been  rightly  chosen. 

The  effect  of  changing  the  standard  temperature  from  32  to  60  degrees 
F.,  and  in  using  higher  instead  of  lower  heating  value  of  the  fuel  is 
easily  found.  From  this  it  is  clear  that 

h  jh 

-    ,    1  -  ev  and  -  •  ev 
a  +  1  a 

do  not  change  with  change  of  standard  temperature,  and  that  at  any 
standard  temperature,  eh  or  eBh  is  constant  for  a  given  fuel  whether 
higher  or  lower  value  is  taken  as  the  standard. 

At  very  high  speeds  the  power  is  limited  by  wire-drawing  as  discussed  in 
Par.  60,  Chap.  XII,  relative  to  steam  engines;  and  in  like  manner  the  range 
over  which  power  is  proportional  to  piston  speed  may  be  increased  by 
large  valves,  free  passages,  proper  valve  setting,  etc.  Maximum  horse- 
power is  reached  at  a  certain  speed,  beyond  which  the  power  decreases. 
This  does  not  mean,  of  course,  that  the  power  is  not  available  at  speeds 
much  beyond  that  giving  maximum  power,  as  the  power  requirements 
may  be  much  less  than  the  maximum,  as  in  automobile  propulsion  with 
good  roads  and  other  favorable  conditions.  Power  and  speed  curves 
are  given  in  Par.  22,  Chap.  V. 

Fuel  Consumption. — From  equations  (12)  to  (14),  Chap.  VIII,  the  fol- 
lowing formulas  are  derived:  let 

WF  =  the  weight  in  pounds  of  liquid  fuel  used  per  hour. 
VF  =  the  volume  in  cubic  feet  at  standard  temperature  and  pressure, 
of  gaseous  fuel  used  per  hour, 

13 


194 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


h  =  the  heating  value  per  pound  of  liquid  fuel  or  per  cubic  feet  of 
gaseous  fuel  at  standard  temperature  and  pressure. 

Then: 

2545# 


eh 


and: 


VF  = 


eh 


(19) 


(20) 


If  H  is  i.h.p.  or  b.h.p.,  e  must  be  the  corresponding  efficiency.  •  If  h 
is  the  higher  or  lower  value,  e  must  be  based  upon  it.  To  find  the  fuel 
per  h.p.-hr.,  divide  by  H. 

77.  Compression. — In  Chap.  VI,  Formulas(ll)  and  (17)  give  the  effi- 
ciencies of  the  constant- volume  and  constant-pressure  cycles  respectively. 
From  these  it  is  apparent  that  the  higher  the  compression  the  greater 


the  efficiency.  There  are,  however,  practical  limits.  In  engines  com- 
pressing the  mixture  of  fuel  and  air,  ignition  will  occur  prematurely  from 
the  heat  generated  thereby  if  compression  is  carried  too  far.  The  igni- 
tion point  varies  with  different  fuels  and  with  different  mixtures.  Within 
certain  limits  compression  may  be  increased  to  advantage  by  employing 
a  lean  mixture — poor  in  fuel;  but  above  a  certain  limit,  the  m.e.p.  de- 
creases too  rapidly,  reducing  the  mechanical  efficiency  as  explained  in 
Chap.  X.  The  theoretical  thermal  efficiency  increases  but  slowly  beyond 
a  certain  increase  in  compression,  and  a  point  is  reached  where  the  product 
of  mechanical  and  thermal  efficiency  will  decrease  if  compression  is 
carried  higher.  This  product,  as  already  shown,  is  the  thermal  efficiency 
at  brake,  or  true  efficiency.  This  is  shown  in  Fig.  109,  taken  from 
Gtildner. 


THE  INTERNAL-COMBUSTION  ENGINE 


195 


It  is  not  well  to  approach  too  near  this  limit.  The  friction  load  is 
increased  by  high  compression  and  unless  there  are  positive  advantages 
to  be  gained,  moderation  should  be  practised.  Engines  with  moderate 
compression  have  sometimes  shown  remarkable  economy,  showing  that 
other  features  of  design  offset  this  seeming  advantage. 

With  the  medium-  and  high-compression  oil  engines,  compression 
is  carried  high  enough  to  insure  complete  combustion.  The  mechanical 
efficiency  of  these  engines  is  low,  but  is  more  than  offset  by  the  attain- 
ment of  high  compression  without  in  any  way  sacrificing  the  charge 
weight. 

In  the  methods  of  power  determination  employed  in  the  preceding 
paragraph  theoretical  efficiency  is  not  considered,  therefore  compression 
pressure  does  not  enter  into  the  discussion.  By  avoiding  extremes, 
it  is  probable  that  considerable  leeway  may  be  allowed  in  the  selection  of 
compression  pressure,  but  in  the  use  of  the  formulas  of  Par.  75  it  is  assumed 
that  a  pressure  suitable  to  the  conditions  of  operation  is  employed. 

The  same  engine  may  sometimes  be  used  for  several  different  fuels, 
but  better  efficiency  or  capacity  might  be  obtained  if  the  compression 
were  changed. 

Table  34  gives  compression  pressures  in  pounds  per  square  inch, 
absolute,  according  to  the  authorities  noted,  while  Table  35  was  given  by 
R.  E.  Mathot  in  Power,  April  11,  1916.  The  m.e.p.'s  in  the  latter  table 
are  maximum  figures  from  well-designed  engines  working  under  favorable 
conditions  with  an  overload  capacity  of  from  10  to  15  per  cent.  Table 
35  gives  values  from  actual  practice  covering  over  600  tests  by  Mr. 
Mathot,  on  about  40  different  makes  of  European  and  American  engines. 

TABLE  34 


Type  of  engine 

Abs.  comp.  pressure 

Authority 

Gasoline 

45  to  95 

Lucke 

Kerosene  (hot  bulb)  .  
Natural  gas  

30  to  75 
75  to  130 

Lucke 
Lucke 

Citv  cas 

60  to  100 

Lucke 

Producer  gas  

100  to  160 

Lucke 

Blast-furnace  gas  

120  to  190 

Lucke 

Natural  gas 

95  to  110 

Roberts 

Illuminating  gas 

105 

Roberts 

Gasoline  

95 

Roberts 

Kerosene        

75 

Roberts 

Blast-furnace  gas 

142 

Roberts 

Producer  gas      

115  to  130 

Roberts 

196 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


TABLE  35 


Kind  of  fuel 

h  (high) 

Comp.  press, 
absolute 

PM 

Blast-furnace  gas  
Producer  gas  .                   

100 
135 

185  to  215 
155  to  185 

70 
75 

Natural  gas 

900 

155  to  165 

90 

Coke-oven  gas  

550 

135  to  145 

80 

Illuminating  gas  (coal  gas)  
Kerosene  ....                                              ... 

650 
20,000 

125  to  155 
55  to    85 

85 
60 

Kerosene  (with  water  injection)  

20,000 

85  to  100 

60 

Benzol  (industrial  engine)  

17,000 

85  to  100 

70 

Benzol  (automobile  engine)                          

17,000 

100  to  115 

90 

Alcohol  (industrial  engine) 

13  000 

115  to  145 

85 

Alcohol  (automobile  engine) 

13  000 

125  to  175 

95 

Crude  oil  (Diesel  engine)                                

18,500 

515 

105 

Crude  oil  (2-cycle  hot  bulb  engine) 

18500 

155  to  175 

45 

Crude  oil  (4-cycle,  hot  bulb  engine)  
Tar  oil  (Diesel  engine)  

18,500 
18,000 

275  to  315 
615 

75 
100 

78.  Governing. — There  are  two  general  methods  employed  to  regulate 
the  speed  of  internal-combustion  engines.  The  first  is  the  intermittent 
impulse  system  in  which  the  working  cycle  is  always  the  same,  but  at 
loads  less  than  the  maximum  the  number  of  cycles  per  minute  is  reduced 
by  skipping.  This  is  known  as  hit-and-miss  governing.  It  is  used 
on  many  small  engines  and  is  economical,  as  the  combustion  of  each 
charge  takes  place  under  the  best  conditions.  Due  to  the  irregularity 
of  its  impulses,  especially  at  light  loads,  it  does  not  regulate  closely; 
this  may  be  in  part  offset  by  a  very  heavy  flywheel,  and  hit-and-miss 
engines  are  sometimes  used  to  operate  small  dynamos,  but  for  larger 
powers  where  close  regulation  is  required,  and  for  driving  alternators 
for  parallel  operation,  they  are  impracticable. 

The  other  general  method  is  the  variable  impulse  system.  In  this 
system  the  m.e.p.  is  changed  as  in  the  steam  engine,  but  no  cycles  are 
skipped.  This  system  has  two  subdivisions:  the  quality  method,  in 
which  the  proportion  of  fuel  to  air  is  varied;  and  the  quantity  method, 
in  which  a  charge  of  constant  quality  is  admitted  during  a  portion  of  the 
suction  stroke,  or  more  commonly,  is  throttled  during  the  entire  stroke. 
Sometimes  a  combination  of  the  quality  and  quantity  methods  is  used. 

The  Quality  Method. — In  this  method  as  applied  to  gas  and  light-oil 
engines,  the  air  inlet  valve  admits  a  full  charge;  the  fuel  supply  is  under 
the  control  of  the  governor  and  is  throttled  at  light  loads.  The  com- 
pression pressure  is  constant,  which  is  considered  a  theoretical  advantage. 


THE  INTERNAL-COMBUSTION  ENGINE 


197 


Fig.  110  is  an  indicator  diagram  from  a  gas  engine  using  quality 
regulation,  the  full  lines  showing  full  load  and  the  dotted  lines  lighter 
loads. 

From  medium  to  full  load  this  method  gives  good  results.  For  very 
light  loads  the  mixture  is  so  lean  as  to  be  difficult  to  ignite  and  slow 
burning,  leading  to  the  possibility  of  skipping  and  poor  economy. 


FIG.  110. — Quality  regulation. 


FIG.  111. 


The  quality  method  is  used  for  the  heavy-oil  engine,  but  in  this,  igni- 
tion depends  upon  the  high  temperature  of  the  highly  compressed  air, 
and  all  fuel  delivered  to  the  cylinder,  no  matter  how  small  the  quantity, 
is  perfectly  consumed.  As  theoretically  indicated  by  Formula  (17), 
Chap.  VI,  the  shorter  the  combustion  line — sometimes  called  the  cut-off — 
the  greater  the  thermal  efficiency,  and  this  is  true  in  practice.  This  is, 
of  course,  the  indicated  efficiency,  the  reduced  mechanical  efficiency  at 
very  light  loads  offsetting  this  gain.  Fig.  Ill  is  an  indicator  diagram 
for  a  Diesel  engine,  the  full  lines  being  for  rated  load  and  the  dotted 
lines  for  a  lighter  load. 

The  Quantity  Method. — This  method  with  variable  admission,  as  usually 
employed,  admits  a  charge  of  uniform 
quality  during  a  portion  of  the  stroke, 
when  the  valve  is  quickly  closed  by  a 
trip  mechanism  similar  in  principle  to 
the  gear  of  a  Corliss  steam  engine. 
Assuming  a  full-load  diagram  similar  to 
Fig.  110,  a  light-load  diagram  (exagger- 
ated for  the  purpose  of  illustration)  is 
shown  in  Fig.  112.  The  charge  is  ad- 
mitted along  the  suction  stroke  from  a  to 
b  when  the  inlet  valve  suddenly  closes.  FlG-  1 12.  — Quantity  method  with 

_.       .  , ,  .      ,  -     . ,  , .  variable  admission. 

During    the    remainder    01    the    suction 

stroke  the  charge  is  expanded  from  b  to  c,  and  theoretically,  at  least, 
passes  over  the  same  curve  on  the  portion  of  the  compression  stroke  from 
c  to  b.  There  is  then  no  fluid-friction  loss  during  this  part  of  the  cycle, 
the  only  loss  of  this  character  being  the  usual  suction  and  back-pressure 


198 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIG.     113.  —  Quantity    method 
throttling  governor. 


with 


losses  caused  by  the  normal  valve  openings  as  shown  by  the  shaded  area 
of  Fig.  112. 

In  the  quantity  method  with  throttling  governor,  a  charge  of  uniform 
quality  is  throttled  throughout  the  suction  stroke.  This  is  best  shown 
by  another  exaggerated  diagram  in  Fig.  113,  assuming  the  same  weight 
of  charge — or  the  same  pressure  at  the  end  of  the  suction  stroke.  The 
fluid-friction  loss  is  indicated  by  the  dotted  area. 

In  reality,  the  shaded  areas  in  both 
Fig.  112  and  Fig.  113  are  relatively 
small,  and  while  the  cut-off  method 
might  be  expected  to  give  slightly 
better  economy,  the  throttling  method 
is  more  often  used,  due  no  doubt  to  the 
simpler  mechanism  required. 

The  usual  argument  against  quantity 
governing  is  the  low  compression  pres- 
sure at  light  loads;  this  tends  to  reduce 
the  rate  of  combustion,  but  with  a 
properly  proportioned  mixture  this  is 
not  so  noticeable. 

Combined  Methods. — By  employing 
as  lean  a  mixture  as  ^will  give  reasonable  economy  at  the  minimum 
load,  and  using  the  quantity  method  between  this  load  and  the  lightest 
load  obtained  with  maximum  compression;  then  if  the  quality  method 
be  employed  for  loads  greater  than  this,  it  is  probable  that  better  results 
may  be  obtained  over  a  wider  range  of  power.  According  to  Power,  this 
method  is  used  with  the  5000  horsepower  gas  engine  of  the  Ford  Motor 
Company. 

The  hit-and-miss  method  might — and  probably  has  been  applied 
to  the  lower  pressure  range,  combined  with  either  the  quality  or  quantity 
method,  and  when  the  closest  regulation  is  not  required  should  give 
good  economy  over  a  wide  range  of  power.  This  arrangement  might 
be  applicable  to  rolling-mill  engines  and  should  give  as  good  regulation, 
surely,  as  the  single-eccentric  Corliss  engines  formerly  used  for  such 
service. 

More  will  be  said  about  hit-and-miss  governing  in  Chap.  XVIII.  The 
apparatus  for  effecting  the  different  methods  of  governing  will  be  ex- 
plained in  Chaps.  XIX  and  XX. 

79.  Rating. — Internal-combustion  engine  rating  is  not  as  uniform  as 
that  of  the  steam  engine.  With  the  larger  gas  engines  and  oil  engines, 
there  is  usually  some  provision  for  overload.  As  previously  stated, 


THE  INTERNAL-COMBUSTION  ENGINE  199 

Giildner  allows  from  15  to  20  per  cent.  A  number  of  manufacturers  of 
Diesel  engines  allow  10  per  cent,  overload  for  a  given  time — such  as  two 
hours,  and  state  that  much  larger  loads  have  been  carried  for  a  limited 
time. 

It  is  apparent  that  some  small  engines  are  rated  at  their  maximum 
capacity,  or  at  the  maximum  load  at  which  they  will  operate  continuously 
without  giving  trouble. 

While  desirable,  it  is  not  essential  to  have  absolute  uniformity  in  rat- 
ing, so  long  as  the  conditions  of  rating  are  plainly  stated.  All  internal- 
combustion  engines  have  a  maximum  capacity  at  which  they  will  run 
continuously  without  heating,  and  in  most  cases  may  exceed  this  capacity 
for  a  limited  time.  This  condition  may  just  fill  the  bill  for  certain  work, 
and  manufacturers  should  determine  these  limits  as  accurately  as  possible 
and  use  them  conservatively  in  their  specifications.  The  minimum  ca- 
pacity of  satisfactory  operation  should  also  be  known.  With  these  data 
at  hand,  and  a  carefu^y  plotted  economy  curve,  the  mill  owner,  con- 
sulting engineer,  and  tho  engine  manufacturer  himself  will  be  able  to 
handle  power  problems  with  accuracy  and  meet  all  guarantees. 

The  lack  of  overload  capacity  is  often  mentioned  as  an  objection  to 
the  internal-combustion  engine,  and  indeed,  when  the  Diesel  engine  is 
rated  to  give  10  per  cent,  overload,  and  it  is  known  that  a  steam  engine 
with  long  range  cut-off  will  run  continuously  and  satisfactorily — barring 
economy — with  an  overload  of  from  50  to  100  per  cent.,  the  flexibility  of 
the  steam  engine  is  certainly  attractive  for  certain  kinds  of  work. 

Take  for  example,  a  200-h.p.  steam  engine  designed  for  50  per  cent, 
overload;  a  200-h.p.  Diesel  engine  as  usually  rated  will  carry  a  maximum 
load  of  220  h.p.,  while  the  steam  engine  will  carry  300  h.p.  Is  this  a  fair 
comparison?  Both  engines  operate  with  maximum  economy  at  200  h.p., 
at  least  approximately.  A  Diesel  engine  rated  at  273  h.p.  would  be 
required  to  develop  a  maximum  power  of  300  h.p.;  then  at  200  h.p., 
the  rated  power  of  the  steam  engine,  the  Diesel  carries  about  73  per  cent, 
of  its  rating.  But  how  does  the  economy  at  this  load  compare  with  the 
economy  of  the  steam  engine  at  the  same  load? 

In  Fig.  114  are  plotted  two  curves  showing  B.t.u.  per  minute  at 
different  powers  for  a  Diesel  engine  and  a  noncondensing,  uniflow  steam 
engine.  Some  assumptions  had  to  be  made  in  reducing  the  data  so 
that  a  comparison  might  be  made,  and  the  chart  may  be  considered  as 
only  suggestive.  If  200  h.p.  is  assumed  as  the  rated  power  of  the  Diesel, 
it  will  not  be  the  most  economical  load,  but  it  will  give  better  economy 
than  the  steam  engine"  at  this  load,  and  will  have  the  same  overload 
capacity 


200 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


If  the  maximum  power  that  may  be  developed  continuously  by  an 
internal-combustion  engine  is  denoted  by  HM  and  the  rated  power  by 
HR,  the  ratio  of  the  maximum  to  the  rated  load  is: 

q  =  ^. 

HR 

The  fraction  of  the  maximum  load  to  give  a  certain  rated  load  is: 

1 


-  -EM  - 


400, 


.350 


50 


100        150       ZOO       250       300      350 
I.H.R 

FIG.  114. 


or: 


The  overload  capacity  in  per  cent,  is : 

Overload  =  100(0  -  1) 

Per  cent,  overload   . 


(21) 


Table  36  gives  the  required  percentage  of  the  maximum  capacity 
required  for  the  rated  load,  to  allow  a  given  overload  capacity. 


TABLE  36 


Overload  in  per  cent,  of  rated  = 
100  (q  -  I).. 

10.0 
91.0 

20.0 
83.3 

30.0 
76.9 

40.0 
71.4 

50.0 
66.6 

60.0 
62.5 

70.0 

58.8 

80.0 
55.5 

90.0 
52.6 

100.0 
50.0 

Rated  load  in  per  cent,  of  maxi- 
mum =  lOO/^  

THE  INTERNAL-COMBUSTION  ENGINE 


201 


Then  with  a  curve  such  as  Fig.  115,  showing  the  fuel  consumption  at 
different  percentages  of  the  maximum  load,  the  rated-load  economy  for 
any  overload  requirement  may  be  determined,  and  guarantees  intelli- 
gently made. 

Any  means  of  increasing  the  economical  load  range,  such  as  suggested 
in  the  preceding  paragraph,  will  increase  the  possibility  of  providing  for 
overloads  with  good  economy  for  the  rated  and  lighter  loads. 

80.  Indicator  Diagrams  and  Maximum  Piston  Thrust. — Actual  indi- 
cator diagrams  are  of  great  value  to  the  designer,  and  some  knowledge 
of  them  absolutely  essential;  but,  as  with  the  steam  engine,  it  is  of  great 
convenience  to  be  able  to  plot  conventional  diagrams  which  approximate 
actual  diagrams,  and  these  may  often  be  used  as  a  basis  of  design. 


tr-o 
|.6 

CQ 

^ 
,§•' 

-o   0 

\ 

X 

-•-. 

^—  1^ 

0     .1      .Z     .3    .4    .5     .6     .7     .8    .9    1.0 
Fraction  of  Max.  Load 
FIG.  115. 

In  Par.  30,  Chap.  VI,  a  method  is  given  of  plotting  diagrams  for  the 
constant-volume  and  constant-pressure  cycles.  PM  may  be  found  from 
Formulas  (1)  or  (2),  or  assumed;  then  if  the  compression  pressure  of  the 
constant-volume  cycle  is  assumed  the  explosion  pressure  may  be  found 
by  a  simple  calculation.  The  compression  pressure  must  also  be  assumed 
for  the  constant-pressure  cycle  in  order  to  determine  the  clearance  volume; 
the  volume  at  the  end  of  combustion  may  then  be  found  by  trial  and 
error. 

The  exponent  of  the  curve  must  be  chosen  so  as  to  give  a  curve  as 
near  the  actual  as  possible ;  then  in  most  cases  with  the  constant- volume 
cycle,  the  clearance  volume  will  be  the  practically  required  volume.  The 
value  of  n  varies  in  practice,  and  is  not  usually  the  same  for  both 
compression  and  expansion  curves,  although  so  assumed  in  order  to 
simplify  calculation.  The  value  is  usually  taken  from  1.3  to  1.35,  but 
is  sometimes  as  high  as  1.5  for  part  of  the  curve.  It  is  not  constant  in 
most  cases  but  an  average  value  may  be  assumed  which  will  give  close 
approximations.  Roberts  gives  1.3  as  the  exponent  of  the  compression 
curve  and  1.35  for  the  expansion  curve.  An  actual  constant-volume 


202 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


diagram  is  given  in  Fig.  116,  with  a  conventional  diagram  giving  the  same 
m.e.p.  The  diagrams  are  drawn  separately  as  part  of  the  outline  coin- 
cides, but  are  to  the  same  scale.  A  pencil  tracing  may  be  made  of  one, 
and  by  laying  it  over  the  other  a  comparison  may  be  made. 

The  m.e.p.  of  the  actual  diagram 
is  112  Ib.  Then  from  (36),  Chap. 
VI,  the  pressure  rise  was  found  to 
be  344  Ib.,  assuming  the  exponent  to 
be  1.3;  the  compression  pressure  is 
100  Ib.  per  square  inch  gage  and  r 
is  found  from  (31),  Chap.  VI.  The 
suction  pressure  was  taken  as  14.7 
Ib.  absolute,  giving  a  ratio  of  clear- 
ance volume  to  volume  of  stroke  of 
0.26  (see  Formula  (33),  Chap.  VI). 

Table  37  gives  ratios  of  clearance 
to  piston  displacement  for  different 
gage  compression  pressures  and  ab- 
solute suction  pressures,  taken  from  a 
FlG-  116<  data    card    by   the    Bruce-Macbeth 

Engine  Co.  It  is  stated  that  14.1  suction  pressure  is  to  be  used  from 
sea  level  to  an  altitude  of  1000  ft.;  13.5  Ib.  from  1000  to  2000  ft.;  13  Ib. 
from  2000  to  3000  ft.,  and  12  Ib.  from  4000  to  5000  ft.  The  value  of  the 
exponent  is  1.3. 

TABLE  37 


Ratio  of  clearance  to  piston  displacement  for  suction  of: 


VJCl^U     JJ1  COQU.J.  C    111    IU. 

14.1 

13.5                           113 

12 

t 

80 

0.302 

0.291 

0.282 

0.264 

90 

0.274 

0.263 

0.255 

0.239 

100 

0.250 

0.241 

0.233 

0.218 

110 

0.231 

0.223 

0.215 

0.202 

120 

0.215 

0.207 

0.200 

0.187 

130 

0.201 

0.194 

0.187 

0.175 

140 

0.189 

0.182 

0.176 

0.165 

150 

0.178 

0.172 

0.166 

0.156 

160 

0.169 

0.163 

0.158 

0.148 

170 

0.161 

0.155 

0.150 

0.141 

180 

0.153 

0.148 

0.143 

0.134 

190 

0.146 

0.141 

0.137 

0.128 

200 

0.140 

0.135 

0.132 

0.123 

210 

0.135 

0.130 

0.126 

0.118 

220 

0.130 

0.125 

0.122 

0.114 

THE  INTERNAL-COMBUSTION  ENGINE 


203 


FIG.   117. 


The  conventional  diagram  of  Fig.  116  was  first  drawn  with  a  vertical 
explosion  line;  then  the  corners  were  rounded,  the  explosion  line  made 
slightly  leaning,  and  from  where  it  meets  the  maximum  pressure  line  a 
new  expansion  curve  is  drawn  with  the  same  exponent.  This  increases 
the  area,  offsetting  the  rounded  corners.  The  modified  diagram  is  shown 
in  heavy  lines. 

If  desired,  the  explosion  line 
maybe  left  vertical  and  connected 
to  the  expansion  line  by  a  curve 
of  small  radius,  and  the  maximum 
pressure  retained  for  safety,  as 
this  diagram  is  mainly  used  for 
computing  the  forces  acting  on 
engine  parts  due  to  gas  pressure. 
Fig.  117  is  a  full-load  diagram 
for  a  Diesel  engine,  and  a  con- 
ventional diagram.  The  m.e.p. 
is  86  lb.,  but  at  this  rating  the  en- 
gine will  carry  an  overload  of  35 
per  cent. 

While  it  is  sometimes  stated  that  the  exponent  of  Diesel  curves  may 
be  as  high  as  1.5,  the  conventional  diagram  of  Fig.  117  has  rectangular 
hyperbolas  for  expansion  and  compression  curves,  and  it  appears  that 
an  exponent  less  than  unity  would  give  a  somewhat  closer  agreement  with 
the  expansion  curve.  Although  this  diagram  agrees  with  the  actual 
diagram  better  than  any  with  an  exponent  greater  than  unity,  the 
equation  of  the  hyperbola  could  not  be  used  to  determine  the  clearance 
volume,  because  the  compression  curve  is  more  like  an  exponential  curve. 
Giildner  says  that  for  usual  loads  the  value  of  c  (Par  30,  Chap.  VI)  is 
2.5,  and  the  ratio  of  the  "cut-off"  (so  called  because  it  appears  like  the 

cut-off  of  a  steam-engine  diagram)  to 
the  entire  stroke  is  0.1.  This  gives 
a  compression  ratio  of  16,  and  with 
a  maximum  compression  pressure  of 
___ _  500  lb.  gage  and  a  suction  pressure 
of  14.7  lb.  absolute,  the  exponent  for 
the  compression  curve  is  1.28.  Using 

this  also  for  the  expansion  curve,  the  m.e.p.  from  (41),  Chap.  VI,  is  119  lb. 
As  this  is  from  a  conventional  diagram  it  must  be  multiplied  by  a  diagram 
factor.  If  this  is  0.88,  the  m.e.p.  is  105  lb.,  a  value  often  found. 

The  actual  compression  curve  of  Fig.  117  is  reproduced  in  Fig.  118 


FIG.   118. 


204  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and    compared    with    a    curve    having    an   exponent    of    1.28,    shown 
dotted. 

For  a  force  diagram  it  is  likely  that  any  of  these  curves  will  give 
accurate  enough  results. 

In  using  (41),  Chap.  VI,  the  value  of  e  may  best  be  found  by  assuming 
some  value  near  2.5  and  solving  for  PM-  If  this  is  greater  than  the  re- 
quired value,  decrease  to  2.4  and  so  on;  a  few  trials  will  give  a  value 
near  enough  for  practical  use.  If  the  hyperbola  is  used,  the  formulas  in 
Par.  56,  with  Fig.  82  of  Chap.  XII  may  be  used,  in  which  x  will  be  zero. 
Maximum  Diagram.  —  In  design,  the  chief  value  of  the  indicator  dia- 
gram is  in  the  determination  of  pressure  at  different  parts  of  the  stroke; 
therefore,  the  maximum  diagram  for  a  given  class  should  be  determined. 
Lucke,  in  his  Gas  Engine  Design  uses  450  Ib.  per  square  inch  gage  as  the 
maximum  pressure  in  the  cylinder;  Giildner  uses  356  Ib.  gage  with  such 
a  factor  of  safety  that  the  stresses  may  not  be  excessive  should  the  pres- 
sure be  425  Ib.  With  the  maximum^  pressure  of  356  Ib.,  Giildner  con- 
structs a  diagram  which  he  uses  in  subsequent  calculations.  The  diagram 
is  for  a  producer-gas  engine  with  a  compression  pressure  of  128  Ib.  gage 
and  m.e.p.  of  96.5  Ib.  This  diagram  is  given  in  Fig.  119  for  comparison. 
The  maximum  pressure  in  the  Diesel  engine  is  predetermined  and  is 
usually  about  500  Ib.  per  square  inch  above  the  atmosphere,  although 

sometimes  as  high  as  600  and  as  low  as 
450  Ib.  To  allow  for  rounding  corners  the 
cut-off  should  be  taken  a  little  later  than 
found  by  the  actual  m.e.p.  In  fact,  with 
both  constant-pressure  and  constant-volume 
diagrams  the  actual  m.e.p.  may  be  divided 
by  a  diagram  factor  before  substituting  in 
(36)  or  (41)  of  Chap.  VI;  the  diagram  factor 
may  be  about  0.9. 

In  cylinder  design  the  maximum  gage 
pressure    must    be    employed.     For    other 
FlG    119  engine     parts    the    maximum    unbalanced 

pressure   is  required;  this  may  include  the 

effect   of   inertia,    but   maximum   stresses  should   also  be  checked  for 
maximum  gas  pressure,  neglecting  inertia. 

Letting  px  denote  the  maximum  unbalanced  pressure  per  square  inch 
with  or  withDUt  inertia  of  the  reciprocating  parts,  and  Px  the  total 
maximum  unbalanced  pressure;  then: 


P,  =  (22) 


THE  INTERNAL-COMBUSTION  ENGINE  205 

For  the  total  unbalanced  pressure  at  any  other  part  of  the  stroke: 

P  =  .'-^  (23) 

This  value  may  be  used  in  checking  some  of  the  combined  stresses 
when  the  maximum  may  not  occur  at  the  point  of  the  maximum  direct 
thrust. 

81.  Application   of   Formulas. — This   will   be   given   by   examples. 

Example  1. — Design  a  single-acting,  4-cylinder,  4-cycle  gas  engine 
using  anthracite  producer  gas,  to  develop  a  rated  b.h.p.  of  500  at  a  piston 
speed  of  750  ft.  per  minute. 

From  Table  30,  h  =  141  B.t.u.,  a  =  1.4  cu.  ft.  (the  maximum)  and 
eB  =  0.24;  and  from  Table  31,  ev  may  be  taken  as  0.82.  From  (9),  for 
one  cylinder: 

r>_oS2n    /          4  X  125  X  2.4  ' 

?8-25Vo24X  0.82X750X141  ~  2L2  m" 
This  may  be  taken  anywhere  from  21  to  22  inches. 

Comparing  with  (15),  taking  K  from  Table  33: 


D  =  0.41  ^5 


700  X  125 


750 
The  engine  size  may  be: 

22"  X  30"  -  150. 

Example  2. — Design  a  single-acting,  3-cylinder,  4-cycle  Diesel  engine 
to  develop  a  rated  b.h.p.  of  300  at  a  piston  speed  of  800  ft.  per  minute. 
From    Table   30,  h  =  18,000,  a  =  323  (maximum)  and  eB  =  0.315; 
and  from  Table  31,  ev  =  0.82. 
Then  from  (12) : 

n         «9*    /~        4  X  100  X323~ 

*8'25  \0.315  X  0.82  X  800  X  18,000  =  16'45  m' 
Comparing  with  (15),  taking  K  from  Table  33: 


D  =  0.41  J 


13,500  X  100 


800 
The  size  may  be  taken  as: 

17"  X  26"  -  185. 

It  is  obvious  that  the  values  of  K  in  Table  33  for  these  two  cases  are 
conservative,  as  they  correspond  with  the  maximum  air  supply  given 
in  Table  30. 

Reference:  The  references  at  the  end  of  Chap.  V  may  be  used  in  con- 
nection with  this  chapter. 


CHAPTER  XV 
THE  STEAM  TURBINE 

82.  Introduction. — Chapter  IV  may  be  considered  as  introductory  to 
the  present  chapter,  in  which  turbine  parts  will  be  treated  only  in  their 
relation  to  capacity  and  economy.  A  working  knowledge  of  Thermo- 
dynamics is  also  essential,  the  flow  of  steam  having  direct  application. 

Aside  from  general  treatment  no  originality  is  claimed.  The  excellent 
works  of  Peabody,  Martin,  Jude  and  Stodola  have  been  studied,  as  well 
as  catalogues  and  other  material  furnished  by  some  of  the  leading  Ameri- 
can manufacturers  of  steam  turbines.  The  prospective  steam  turbine 
designer  is  advised  to  fill  in  the  outline  here  given  by  a  study  of  the 
works  mentioned. 

Notation. 

A  =  the  reciprocal  of  Joule's  equivalent  =  M?8- 
k  =  the   thickness   factor — the   fraction    of   the   arc    containing 
nozzles,  which  is  available  after  deducting  blade  thickness, 
etc. 

m  =  the  ratio  of  blade  length  to  pitch  diameter  of  wheel  or  dia- 
phragm. 

y  =  fraction  of  kinetic  energy  used  to  overcome  friction. 
q  =  velocity  coefficient — the  ratio  of  actual  velocity  to  what  it 

would  be  without  friction. 
z  —  fraction  of  residual  energy  available  for  kinetic  energy  in  next 

nozzles, 
/i  =  fraction  of  increase  of  kinetic  energy  due  to  conservation  of 

residual  energy. 
A  =  change  of. 

i  =  fraction  of  heat  drop  per  stage  utilized  in  reaction  turbine. 
j  =  fraction  of  exit  energy  from  blade  or  guide  of  a  reaction  tur- 
bine utilized  to  produce  kinetic  energy  in  next  guide  or  blade. 
e  =  "indicated"  thermal  efficiency. 
eM  =  mechanical  efficiency. 
eD  =  diagram  efficiency. 
eB  —  blade  efficiency  of  a  reaction  turbine. 

206 


THE  STEAM  TURBINE  207 

F  =  heat  factor — ratio  of  actual  " indicated"  thermal  efficiency  to 
Rankine  efficiency.     Sometimes  referred  to  as  " total"  heat 
factor  in  multistage  turbines.     The  "overall"  heat  factor  is 
equal  to  eMF,  in  which  eM  may  include  turbine  and  generator. 
Fs  =  heat  factor  per  stage,  or  hydraulic  efficiency. 
FR  =  reheat  factor  =  F/FS. 
FD  =  distribution  factor. 
R  =  distribution  ratio. 
V  =  velocity  in  feet  per  second  in  general;  also  denotes  flow  from 

nozzles  or  guides. 

VN  =  relative  velocity  of  entrance. 
Vx  =  relative  velocity  of  exit. 
VR  =  residual  velocity. 
Vw  =  velocity  of  whirl. 
VF  =  velocity  of  flow. 

S  =  velocity  at  pitch  line  of  blades  in  feet  per  second. 
W  =  steam  consumption  in  pounds  per  horsepower-hour. 
G  =  total  steam  consumption  in  pounds  per  hour. 
w  =  weight  of  steam  in  pounds  per  second.    , 
a  =  nozzle  area  in  square  inches. 
v  =  specific  volume  of  steam  of  any  condition. 
P  =  pressure  in  pounds  per  square  inch  absolute. 
fw  =  tangential  force  in  pounds  acting  upon  blades. 
fF  =  axial  thrust  upon  blades  in  pounds. 
E.  =  energy  in  foot-pounds. 
t  =  temperature  in  degrees  F. 
C  =  heat  content  of  steam  in  B.t.u.  above  32  degrees  F.,  for  any 

condition.  •{ 

AC  =  adiabatic  heat  drop;  any  portion  of  Ci-C2    along  the  same 

entropy  as  that  of  Ci. 
h  =  heat  of  the  liquid. 
L  =  latent  heat  of  steam. 
x  =  dryness  fraction. 
H  =  turbine  horsepower. 
kw  =  kilowatts. 

D  =  diameter  of  pitch  circle  of  wheel  in  inches. 
d  =  radial  length  of  blade  in  inches. 
c  —  tip  clearance  in  inches. 
nB  =  number  of  rows  of  moving  blades. 
nv  =  number  of  velocity  stages. 
nP  —  number  of  pressure  stages. 


208  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

N  =  r.p.m. 

a  =  angle  of  nozzle  or  exit  of  guide. 
6  =  angle  of  entrance  to  moving  blade. 
0  =  angle  of  exit  from  moving  blade.     Also  entropy. 
5  =  angle  due  to  flow  of  residual  steam;  also  angle  of  entrance 
to  guide. 

83.  The  General  Method. — A  critical  study  of  steam  flow  through  a 
turbine,  other  than  that  based  upon  rather  broad  general  principles, 
involves  problems  of  great  complexity  and  will  not  be  attempted  in  this 
book.  A  few  simple  principles  combined  with  the  broader  and  better- 
known  coefficients  of  performance,  and  a  few  of  the  simpler  refinements 
to  give  a  better  understanding  of  the  principles  involved,  will  be  the  basis 
of  procedure. 

If  in  determining  the  power  of  a  steam  turbine  a  method  parallel  to 
that  of  the  steam  engine  were  employed,  that  is:  if  the  sum  of  the  work 
done  upon  each  blade  were  determined  in  detail,  there  would  enter  into 
the  calculation  so  many  uncertainties  that  the  accumulation  of  errors 
would  render  the  determination  useless.  It  is  obvious  that  the  large 
surface  of  the  engine  piston  acted  upon  by  accurately  known  steam  pres- 
sures offers  a  different  problem  than  the  turbine  with  its  multitude  of 
blades,  even  though  the  mechanical  principles  involved  in  the  latter  are 
simple.  A  more  general  method  is  therefore  best  adapted  to  the  turbine 
and  will  be  outlined. 

As  it  is  impossible  to  take  indicator  diagrams  from  a  turbine,  there  can 
be  no  indicated  horsepower.  An  equivalent  to  this,  the  power  developed 
by  the  action  of  the  steam  upon  the  blades,  is  called  the  turbine  horsepower 
and  will  be  denoted  by  H.  The  thermal  efficiency  is  denoted  by  e; 
C  is  the  heat  content  for  steam  of  any  condition  and  h  the  heat  of  the 
liquid  as  in  Chap.  VII.  The  steam  consumption  per  hour  is  denoted  by  G 
and  the  consumption  per  h.p.-hr.  by  W. 

Letting  subscript  1  refer  to  initial  absolute  pressure  and  2  to  absolute 
exhaust  pressure — being  that  of  the  atmosphere  or  condenser — Equation 
(4),  Chap.  VIII  may  be  written: 

W  _       2545 
«(Ci-fe) 

From  (9),  Chap.  VIII,  and  (8),  Chap.  VII: 

e  =  F^=-^  (2) 

GI    —    ft2 

where  F  is  the  heat  factor,  or  the  ratio  of  actual  "indicated"  efficiency 
to  that  of  the  Rankine  cycle. 


THE  STEAM  TURBINE  209 

From  (1)  and  (2) : 

W_G_    _2545_ 

"  H      F(d  -  d) 
Or: 

TT        Crr  (O  i         ^2/  //i\ 

^54^ 

This  formula  also  applies  to  the  steam  engine  but  is  not  of  practical 
use  in  power  determination.  Ci  —  C2  is  always  the  expression  for  heat 
drop  due  to  adiabatic  expansion,  therefore  Ci  and  C2  must  always  have 
the  same  entropy. 

The  heat  factor  F,  sometimes  called  the  efficiency  ratio,  is  the  product 
of  several  factors,  such  as  blade  efficiency  due  to  form,  and  factors  due 
to  friction  of  nozzles,  blades  and  discs,  and  to  radiation  losses.  Experi- 
ment and  experience  have  fixed  some  of  these  factors  approximately, 
but  their  product  must  always  check  with  F  as  found  from  test;  the  value 
of  F  found  from  tests  is  therefore  more  reliable. 

As  many  steam  turbines  are  employed  for  driving  electric  generators, 
they  are  often  rated  in  kilowatts;  all  calculations  in  this  book  will  be 
made  in  turbine  horsepower  which  is  obtained  from  kilowatt  rating  thus: 

(5) 


eM 

where  eM  is  the  mechanical  efficiency  of  turbine  and  generator. 
Steam  consumption  per  kw.-hr.  is: 

G^  =  1.34TF  (6) 

kw          eM 

As  complete  expansion  is  always  assumed  in  turbine  operation,  F 
must  be  based  upon  the  Rankine  cycle  with  complete  expansion,  given 
by  (10),  Chap.  VIII,  which  is: 

2545  (. 

=  W(d  -  C2) 
Equating  (4)  and  (5)  gives: 

3410       kw 
=  d  -  C2 '  G  ' 
This  is  sometimes  known  as  the  "overall"  heat  factor. 

The  heat  factor  is  fairly  well  known  for  turbines  of  given  power,  and 
ranges  from  0.45  to  0.75,  and  may  be  higher. 
Formula  (3)  may  be  written: 

2545ff  ,„, 

°  -  F(d  -  C.) 
Assuming  F,  finding  Ci  and  C2  from  entropy  table  or  chart  for  the  selected 

14 


210  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

pressures  and  quality,  (8)  gives  the  weight  of  steam  per  hour  which  must 
be  passed  through  a  turbine  of  given  power.  It  now  remains  to  propor- 
tion the  various  turbine  parts  to  handle  this  steam  with  a  minimum  loss. 
84.  Nozzles  and  Other  Passages.  —  The  general  formula  for  friction- 
less,  adiabatic  flow  of  one  pound  of  steam  is  given  by  (12),  Chap.  VII,  and 
is: 


=  223.7       C-C2  (9) 

where  V  is  the  velocity  in  feet  per  second  when  the  initial  velocity  is  zero. 
The  heat  content  C  may  be  calculated  from  Formulas  (2),  (3)  and  (4)  of 
Chap.  VII,  for  dry  saturated,  wet,  and  superheated  steam  respectively, 
with  the  help  of  an  ordinary  steam  table;  but  may  be  more  conveniently 
taken  from  Peabody's  entropy  table  or  a  Molier  diagram. 

In  Equation  (9)  and  in  what  follows,  the  subscript  2  does  not  neces- 
sarily refer  to  the  exhaust  pressure  as  assumed  in  Par.  83,  but  may  indicate 
any  pressure  less  than  pi  at  which  calculations  are  required. 

The  weight  of  steam  in  pounds  per  second  which  will  flow  through  an 
orifice  is,  taking  the  area  of  the  orifice  in  square  inches: 

' 


The  specific  volume  v  may  be  found  in  Peabody's  entropy  table.  Both 
V  and  v  are  the  values  corresponding  to  the  area  considered. 

It  may  be  shown  mathematically  that  when  a  maximum  weight 
of  gas  passes  through  an  orifice  there  is  a  certain  definite  ratio  between 
the  initial  pressure  and  the  pressure  in  the  orifice;  and  that  the  latter 
does  not  decrease  nor  the  rate  of  flow  increase,  however  much  the  pressure 
against  which  the  orifice  discharges  is  lowered.  The  calculated  ratio  for 
air  is  0.5274.  Experiments  on  saturated  steam  give  a  value  of  about 
0.58,  sometimes  taken  as  0.6. 

There  are  a  number  of  empirical  formulas  for  flow  through  an  orifice 
and  the  agreement  with  (9)  is  very  close.  These  formulas  may  be 
found  in  engineering  handbooks  and  will  not  be  given  here. 

Formula  (10)  may  be  written: 

(11) 


It  is  well  known  that  if  any  fluid  flows  at  high  velocity  through  an  orifice 
with  sharp  edges  as  in  Fig.  120,  the  cross-sectional  area  of  the  jet  will  be 
less  than  the  area  of  the  orifice,  forming  a  vena  contracta.  If  the  orifice 


THE  STEAM  TURBINE 


211 


is  formed  as  in  Fig.  121  the  jet  will  fill  the  orifice,  in  which  case  Formula 
(11)  applies,  and  indicates  that  the  velocity  increases  at  a  higher  rate  than 
the  volume,  the  ratio  being  higher  at  the  entrance.  Fig.  121  is  known 
as  a  converging  nozzle  and  is  in  general  the  form  used  for  the  entrance  of 


FIG.  120.  FIG.  121. 

all  machined  nozzles.  Nozzles  in  lower  stages  of  compound  turbines, 
sometimes  called  guide  vanes,  are  often  formed  by  "casting  in"  steel 
plates  as  in  Fig.  122.  Assuming  the  radial  width  uniform,  the  nozzle 
converges  whether  1-2  or  1-3  be  considered  the  width  of  entrance.  The 


I 


FIG.  122. 

distance  4-5  is  the  minimum  width  and  this  may  continue  a  short  distance 
to  6-7 ;  if  this  distance  is  made  too  great,  frictional  loss  will  result.  The 
exit  of  the  nozzle  is  at  6-7,  the  jet  being  formed  by  the  time  it  reaches  this 
cross  section;  and  if  the  pressure  against  which  it  flows  is  not  less  than 
0.58  of  the  pressure  at  entrance,  it  will  retain 
its  sectional  form,  dimensions  and  direction 
after  it  leaves  the  nozzle. 

Neglecting  friction,  the  velocity  of  flow  will 
be  that  given  by  (9),  the  heat  content  C2 
being  that  obtained  at  a  pressure  0.58  of  the 
absolute  pressure  of  the  entering  steam. 

If  the  pressure  against  which  the  jet  flows  is 
appreciable  less  than  0.58  of  the  initial  pressure,  it  will  have  a  tendency 
to  scatter  and  its  kinetic  energy  will  be  dissipated  if  a  converging  nozzle 
is  used,  this  effect  being  more  marked  the  lower  the  discharge  pressure; 
this  makes  it  useless  for  turbine  propulsion.  Adding  a  diverging  passage 
to  the  converging  nozzle  as  in  Fig.  123  overcomes  this  difficulty  by  con- 


FIG.  123. 


212  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

trolling  the  direction  of  flow  until  the  pressure  and  specific  volume  cor- 
respond to  the  pressure  against  which  the  nozzle  discharges;  the  result 
is  a  compact  jet  of  the  same  cross-section  as  the  nozzle  exit.  From  (11), 
this  indicates  that  in  the  diverging  portion  the  specific  volume  increases 
more  rapidly  than  the  velocity.  The  relative  rate  of  this  increase  has  been 
the  subject  of  theoretical  speculation,  and  curves  of  various  form  have 
been  tried,  connecting  the  smallest  area,  or  throat  to  the  exit;  but  the 
straight  line,  commercially  the  simplest  and  cheapest,  seems  to  give 
practically  as  good  results  as  any  other  and  is  generally  used. 

The  throat  area  is  determined  as  for  the  exit  of  the  converging  nozzle, 
taking  C  and  v  corresponding  to  0.58  of  the  initial  pressure;  the  exit  area 
is  found  by  using  C  and  v  for  the  exit  pressure.  This  calculation  may  be 
made  in  two  stages,  using  the  throat  pressure  for  the  initial  pressure  of 
the  diverging  portion,  and  adding  the  velocity  so  obtained  to  the  throat 
velocity,  but  it  is  not  so  satisfactory. 

Example.  —  Design  a  converging-diverging  nozzle  to  expand  adiabat- 
ically  y±  Ib.  of  steam  per  sec.  at  150  Ib.  gage,  to  atmospheric  pressure. 

Taking  the  nearest  absolute  pressures  from  Peabody's  entropy  table, 
Ci  is  1193.3  for  an  absolute  pressure  of  164.8  (entropy  1.56).  At  96.1 
Ib.  absolute—  approximately  0.58  of  164.8—  C2  is  1149.8.  Then  from  (9) 
the  velocity  at  throat  is: 

V  =  223.7\/43^5  =  1476  ft.  per  sec. 

The  specific  volume  at  throat  from  entropy  table  is  4.412.  Then 
from  (11): 

144  X0.25  X  4.412  AQ 

a  =  1476  =  °-108  sq'  m' 

The  corresponding  diameter  for  a  round  section  is  nearly  %  in. 

At  exit,  C2  =  1018  at  14.7  Ib.  absolute  and  entropy  1.56.  Then  from 
(9): 

V  =  223.7\/175.3  =  2960ft.  per  sec. 

At  14.7,  v  =  23.13,  and  from  (11): 

144  X  0.25  X  23.13 

"296^  =  °'281  Sq'  in' 

For  a  round  nozzle  this  is  between  !%2  and  /-g  m-  diameter. 

If  the  nozzle  is  for  a  condensing  turbine  with  28-in.  vacuum,  the  exit 
pressure  nearest  to  this  in  the  table  is  1.005.  €2  =  871.1. 

Then  from  (9)  : 


V  =  223.7  \3222.  =  4015  ft.  per  sec. 


THE  STEAM  TURBINE 


213 


At  this  pressure,  v  =  256.8,  and  from  (11): 
144  X  0.25  X  256.8 


4015 


=  2.31  sq.  in. 


For  a  round  nozzle  this  is  between  ll^{Q  and  1%  in.  diameter. 

Divergent  nozzles  formed  by  cast-in  plates  are  like  Fig.  122,  but  the 
portion  between  4-5  and  6-7 
is  made  longer;  the  area  at  4-5 
is  the  throat,  and  the  diverg- 
ence is  made  by  increasing  the 
radial  width  at  the  exit  6-7. 
This    is   usually    done    in   a" 
straight    line.     The    form   is 
not    one    of    great    accuracy 

and  the  sides  formed  by  the  casting  are  rough.  Fig.  124  gives  some 
idea  of  the  general  form. 

In  Formula  (11),  a  may  be  taken  as  the  total  nozzle  area,  being  the 
product  of  the  number  of  nozzles  and  the  area  of  a  single  nozzle.  For 
nozzles  of  the  type  shown  in  Figs.  122  and  124,  it  is  often  more  convenient 
to  take  the  total  effective  opening  in  the  diaphragm.  In  Fig.  125  let  I 
be  the  length,  measured  at  the  pitch  line,  of  the  opening  containing  cast-in 
plates,  forming  nozzles.  This  is  the  entire  circumference  in  some  cases. 
The  effective  area  of  exit  of  each  nozzle  is  bd,  and  b  is  equal  to  61  sin  a. 
Let: 

Z&i  =  kl 

where  k  is  less  than  unity  and  allows  for  the  thickness  of  the  plate;  k 
may  be  t/(b  +  t),  or  it  may  also  allow  for  bridges  used  to  strengthen  the 
diaphragm  when  the  nozzles  extend  around  the  entire  circumference. 
Then  the  effective  area  with  all  dimensions  in  inches,  is: 

a  =  kdl  •  sin  a  (12) 

This   also  applies  at  the  throat  by 
using  d'  (Fig.  124)  in  place  of  d. 

If  I  =  irD,  where  D  is  the  diameter 
of  nozzle  pitch  circle  in  inches,  and  the 
ratio  d/D  be  denoted  by  m,  then  (12) 
becomes: 


FIG.  125. 


a  =  irkmD2  -  sin  a 


(13) 


Effect  of  Friction. — In  the  form  of  nozzle  shown  in  Fig.  121,  Prof. 
Rateau  found  the  actual  flow  practically  equal  to  the  theoretical  when 
the  exit  pressure  was  0.58  of  the  initial ;  but  when  discharging  into  higher 


214  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

pressures  than  this  the  flow  decreased,  the  discrepancy  between  theoret- 
ical and  actual  being  greater  the  less  the  pressure  drop.  Martin  (Design 
and  Construction  of  Steam  Turbines)  says  that  this  was  probably  due 
to  the  formation  of  a  vena  contracta,  reducing  the  effective  area. 

In  diverging  nozzles,  the  increased  surface  causes  friction,  which 
retards  the  flow.  Various  factors  combine  to  make  the  actual  discharge 
less  than  the  theoretical,  and  it  is  well  to  provide  for  this.  Let  y  be  the 
fraction  of  the  theoretically  available  heat  energy  dissipated  by  friction 
and  radiation;  also  let  it  include  any  loss  due  to  vena  contracta  or  any 
influence  resulting  from  an  improper  form  of  passage,  since  it  is  probably 
true  in  part  that  a  correction  of  one  will  .also  reduce  the  other.  Then 
the  fraction  of  the  heat  drop  available  is  1  —  y,  and  the  equation  for 
flow  is  given  by  (13),  Chap.  VII,  which  is: 


=  223.7      (l  -y)(Ci  -C2)  (14) 

The  expression  1  —  y  is  known  as  the  nozzle  efficiency. 

If  the  loss  were  wholly  due  to  friction  (which,  for  the  usual  com- 
mercial nozzles  is  probably  an  accurate  enough  assumption  for  the  present 
purpose),  a  quantity  of  heat  y(C±  —  C2)  would  be  returned  to  the  steam 
during  its  flow  through  the  nozzle,  increasing  its  entropy  and  volume. 
Formulas  (14),  (18)  and  (20)  of  Chap.  VII  give  the  resulting  quality 
of  steam  at  pressure  p2  when  the  heat  quantity  y(Ci  —  C2)  is  added 
to  the  heat  content  resulting  from  adiabatic  expansion  to  that  pressure. 
The  specific  volume  for  this  new  value  may  be  substituted  in  (11)  and 
the  area  determined.  The  entropy  table  may  be  readily  used  for  such 
problems. 

Example. — Using  the  same  data  as  in  the  previous  example,  determine 
throat  and  exit  areas  when  the  values  of  y  are  0.05  and  0.15  respectively. 

From  the  entropy  table  Ci  —  C2  for  the  throat  is  43.5  as  before;  then 

from  (14) :  

V  =  223.7  V0.95  X  4375  =1437. 

The  resulting  heat  content  at  96.1  Ib.  is: 

C2  +  y(Ci-  C2)  =  1149.8  +  2.175  =  1151.975  B.t.u. 

This  occurs  between  the  entropy  1.56  and  1.57  and  we  must  interpolate 
to  find  the  specific  volume  as  follows: 

1157.7  1151.97     4.451  4.4120 

1149.8  1149.80     4.412  0.0107 
7.9  is  to     2.17  as   0.039  is  to  0.0107    v  =  4.4227 


THE  STEAM  TURBINE  215 

Then  from  (11): 

144  X  0.25  X  4.4227 

a  =  1437  =  o-111  sq- m- 

This  is  less  than  3  per  cent,  greater  than  before;  the  increase  due  to 
drying  the  steam  is  about  0.5  per  cent. 
At  exit: 

V  =  223.7  V0.85  X  175.3  =  2750 

The  resulting  heat  content  at  14.7  Ib.  is: 

1018  +  26.3  =  1044.3  B.t.u. 

The  heat  content  at  entropy  1.6  is  1044.9,  which  will  be  assumed  as  close 
enough.  Then  v  =  23.88,  and  from  (11): 

144  X  0.25  X  23.88 

-2750-  =  0.312  sq.  in. 

This  is  an  increase  of  11  per  cent,  over  the  area  found  by  neglecting 
friction;  about  3.3  per  cent,  of  this  is  due  to  increased  specific  volume, 
the  remainder  to  decreased  velocity. 

Using  ordinary  steam  tables  for  determining  v,  XN,  the  new  dryness 
factor  due  to  the  degradation  of  heat  may  be  found  as  follows:  The 
value  of  #2  at  14.7  Ib.  due  to  adiabatic  expansion  from  dry  steam,  using 
the  nearest  value  of  initial  pressure  to  164.8,  may  be  found  by  Formulas 
(5)  and  (6),  Chap.  VII,  and  is: 

0.524  +  1.037  -  0.3118 
-L4447- 

Then  from  (14),  Chap.  VII: 

.,-  0.865  °^P-3  =  0.892 
From  (16),  Chap.  VII: 

v2  =  0.017  +  (0.892  X  26.773)  =  23.817  cu.  ft. 

as  compared  with  23.88  found  by  the  entropy  table.  All  calculations 
were  performed  on  the  slide  rule,  and  with  the  exception  of  the  determina- 
tion of  volume  at  throat  pressure  from  the  entropy  table,  no  interpolations 
were  made. 

The  ratio  of  velocity  reduced  by  friction  to  that  due  to  adiabatic 
expansion  is: 

1  ~  y  =  VU85  =  0.922. 


216 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Formula  (21),  Chap.  VII  gives  the  heat  returned  to  the  steam  in 
terms  of  this  velocity  ratio,  which  is  there  denoted  by  q}  and  is: 

Heat  returned  =  296Q2,(n1  ~  °'85)  =  26.3 

oUjUUU 

which  is  exactly  the  value  found  before. 

The  factor  q  is  known  as  the  velocity  coefficient  and  its  relation  to  y 
is: 

q  =  Vl-y  (15) 

or: 

V  =  1  -  q2  (16) 

Other  Passages    Used  as  Nozzles. — The  fixed  vanes,  or  guides,  of  a 


FIG.  12G. 

reaction  turbine  serve  as  convergent  nozzles  and  are  shown  in  Fig.  126, 
from  Martin.  This  design  is  called  the  normal  Parsons  blading,  and  the 
shortest  distance  between  the  blades  is  made  equal  to  %  the  pitch.  Ac- 
cording to  Martin  (Design  and  Construction  of  Steam  Turbines),  the 
discharge  angle  varies  as  the  pitch  is  changed  and  cannot  be  determined 
by  the  tangent  to  the  curve  on  the  concave  side,  as  the  back  of  the  adja- 
cent blade  is  also  curved  and  has  a  different  tangent.  In  blades  %  in. 
wide  (axial  width),  the  pitch  of  the  blades  in  the  casing  (guides)  is  Y±  in. 
and  on  the  rotor  the  pitch  is  ^f  g  in.  By  experiment,  the  discharge  angle 
of  the  guides  is  between  17  and  18  degrees,  and  for  the  blades  on  the 
rotor,  between  18  and  19  degrees.  Another  form  of  blade,  called  the 
wing  blade  is  used  where  large  discharge  angles  are  required  and  is  shown 


THE  STEAM  TURBINE 


217 


in  Fig.  127.  The  discharge  angle  of  wing  blades  is  from  40  to  50  degrees. 
Semi-wing  blades  are  normal  blades  specially  spaced  and  set  to  have  a 
discharge  angle  of  from  28  to  30  degrees. 

Due  to  the  rounded  backs  of  Parsons  blades,  the  stream  lines  unite 
as  shown  by  the  dotted  lines  of  Fig.  126,  and  according  to  Martin,  no 


i.    ^ 


FIG.  127. 


deduction  is  needed  for  thickness.  Then,  as  these  guides  extend  around 
the  entire  periphery,  k  in  (12)  and  (13)  is  unity.  Peabody  says  the 
necessary  allowance  for  blade  thickness  is  indefinite. 

Guide  blades  of  velocity-stage  impulse  turbines  are  used  to  change 
the  direction  of  steam  at  constant  pressure  and  are  not  considered  as 
nozzles.  The  simplest  form  of  guide  is  symmetrical — with  inlet  and  dis- 


B  C 

FIG.  128. 

charge  angles  equal,  as  in  Fig.  128.  The  width  of  the  passage  is  constant 
and  neglecting  friction,  the  radial  height  d  is  constant.  If  friction  is 
considered,  there  will  be  a  slight  increase  in  radial  height  on  the  discharge 
side  as  shown  in  Fig.  128(7.  Assuming  q  the  velocity  coefficient,  the  heat 
returned  by  friction  is  given  by  (21),  Chap.  VII.  Putting  this  in  place  of 
y(Ci  —  (72)  in  (14),  Chap.  VII  (assuming  wet  steam),  gives  a^;  then  find  v 


218 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


from  (16),  Chap.  VII  and  substitute  in  (11)  of  this  chapter,  using  qV0  for 
Fo,  the  relative  velocity  at  entrance.  If  the  subscript  N  refers  to  the 
increased  area: 

a*  _  v N  _  #A- 
a       qv       qx 


(17) 


as,  neglecting  the  volume  of  water  in  the  steam,  the  volume  is  directly 
proportional  to  the  dryness  factor.     Then : 

(18) 


x  50,OOOzL 

where  L  is  the  latent  heat  at  the  pressure  in  the  guide.  By  similar  sub- 
stitution, conditions  for  superheated  steam  may  be  found  from  (18)  and 
(20),  Chap.  VII,  the  latter  being  for  initially  wet  steam  becoming  super- 
heated. When  the  entropy  table  (Peabody's)  is  used  a  more  direct  way 


FIG.   129. 


may  be  to  add  the  heat  returned  as  given  by  (21),  Chap.  VII  to  the  heat 
content  at  the  pressure  considered,  and  then  find  x.    The  heat  returned  is : 


QR  = 


TV(1  -  i 
50,000 


Unsymmetrical  guides  are  shown  in  Fig.  129.  In  this  case  the  passage 
becomes  narrower  toward  the  exit,  and  for  constant  area  of  passage,  the 
height  d  must  increase  so  that  the  product  of  width  and  height  is  constant. 
This  is  approximated  by  making: 


or: 


61 

r~ 

62 


=  M2 

_  sin  A 
sin  B 


THE  STEAM  TURBINE  219 

Then  the  connecting  line  is  made  straight.     This  must  now  be  multiplied 
by  (17)  if  friction  is  to  be  considered.     Then: 

,        XN  bi     ,        a  AT  sin  A 

d>2  =  —  •  =-  •  «i  = ; — 5  •  ai  (19) 

qx  62  qx  sin  B 

In  some  cases  the  ratio  dz/di,  as  given  by  (19)  is  excessive  and  would 
lead  to  impracticable  blade  dimensions;  this  may  be  remedied  by  in- 
creasing di  after  dz  has  been  determined.  This  is  also  discussed  at  the 
end  of  Par.  88.  In  applying  (19)  the  angle  A  should  be  the  angle  deter- 
mined by  the  velocity  diagram  and  not  the  actual  blade  angle  if  increased 
as  in  Fig.  135  (to  be  explained  later). 

These  equations  may  also  apply  to  moving  blades  of  impulse  turbines 
when  q  refers  to  relative  velocity. 

The  form  of  guide  in  Fig.  129  is  sometimes  used  for  convergent 
nozzles,  in  which  case  d%  =  di  and  the  area  bzd2  must  be  determined 
from  (11)  as  for  any  nozzle. 

85.  Practical  Notes  on  Nozzles  and  Other  Passages. — Nozzles  should 
be  used  only  with  the  range  of  pressure  for  which  they  are  designed. 
Over-expansion,  or  carrying  the  expansion  in  the  nozzle  to  a  pressure  below 
that  into  which  the  steam  flows,  causes  a  disturbance  in  the  nozzle,  with 
a  loss  of  efficiency  which  increases  rapidly  with  increase  of  back  pressure. 
Experiments  have  shown  that  a  slight  amount  of  under-expansion  in  the 
nozzle — to  a  pressure  slightly  greater  than  the  back  pressure — is  benefi- 
cial to  blade  efficiency.  This  may  be  accomplished  by  slightly  reducing 
the  exit  area;  possibly  the  neglect  of  the  increase  of  specific  volume 
due  to  friction  would  provide  the  right  allowance,  in  which  case  all 
formulas  for  nozzle  design  are  contained  in  this  paragraph,  and  no 
reference  to  Chap.  VII  need  be  made.  The  decreased  velocity  given  by 
(14)  must  not  be  overlooked. 

Martin  says  that  the  entrance  to  the  throat  of  a  nozzle  must  be  "very 
easy  and  well  rounded"  (referring  to  a  round  nozzle)  or  the  discharge 
may  be  only  0.8  or  0.9  of  the  theoretical.  Jude  says  that  the  inlet  end 
should  not  have  a  large  radius,  but  a  small  rounding  off  is  advantageous. 
He  further  states  that  a  round  nozzle  reasonably  correct  in  shape,  and 
not  working  with  pressures  entirely  unsuitable,  should  give  a  velocity 
coefficient  q  of  0.95,  but  that  undoubtedly  q  is  a  little  lower  for  square  or 
rectangular  nozzles  "on  account  of  the  natural  internal  instability  of  the 
jet."  Also,  "an  appendage  to  a  circular  nozzle,  to  change  the  circular 
jet  into  a  square  or  rectangular  jet,  also  involves  a  loss  of  efficiency,  rarely 
less  than  another  3  per  cent,  (velocity),  and  may  often  amount  to  10  or 
15  per  cent.,  according  to  the  way  it  is  made,  its  continuity  and  the 
condition  of  the  surfaces. " 


220  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Martin  says  that  curvature  has  an  adverse  effect  on  efficiency,  es- 
pecially when  the  curves  leading  to  the  throat  are  abrupt. 

For  experiments  upon  straight  convergent-divergent  nozzles  with 
easy  well-rounded  entrance  and  taper  not  too  rapid,  Martin  gives  a 
formula  for  loss  of  energy,  which  in  our  notation  is: 

6(Cl_C2_45) 
10,000 

He  says  that  experiments  upon  nozzles  with  curved  entrance  and  in 
which  a  round  section  changed  to  rectangular  (nearly  square)  at  exit, 
gave  values  fully  twice  as  great.  For  the  example  of  nozzle  design  pre- 
viously given,  Ci  -  C2  =  175.3.  Then  from  (20),  y  =  0.078,  or  1  -  y 
=  0.922.  From  (15),  q  =  0.96;  or  the  loss  of  velocity  is  4  per  cent. 
Twice  the  energy  loss  gives  1  —  y  =  0.844,  or  q  =  0.918;  or  the  velocity 
loss  is  8.2  per  cent.  These  figures  may  be  compared  with  those  given 
by  Jude. 

Martin  says  that  round  nozzles  lead  to  inefficiency  in  turbine  practice. 
This  is  probably  due  to  the  entrance  of  a  circular  jet  into  the  rectangular 
opening  between  the  blades.  However,  round  nozzles  are  much  used, 
especially  in  the  first  stages  of  certain  types. 

No  definite  rules  can  be  given  for  taper  of  nozzles.  Too  small  a 
taper  gives  long  nozzles  with  resulting  friction;  while  if  too  large,  the 
effect  upon  the  jet  is  detrimental.  A  total  taper  (change  of  diameter) 
of  from  1  in  6  to  1  in  12  is  used  for  straight  round  nozzles.  A  smaller 
taper  is  sometimes  used,  especially  when  the  increase  of  area  in  the  diver- 
gent portion  is  small.  Practical  construction  may  sometimes  call  for  a 
much  .greater  taper  than  1  in  6  for  such  nozzles  as  shown  in  Fig.  124, 
and  in  all  cases  judgment  must  be  used. 

From  various  experiments,  Martin  gives  some  practical  values  of  y 
and  q  as  follows:  When  V  =  800  to  1200,  q  =  0.92  and  y  =  0.15.  For 
nozzles  similar  to  Figs.  122  and  124,  the  values  were  lower.  The  maxi- 
mum values  found  for  these  were  with  a  superheat  of  130  degrees  F.  and 
with  V  =  1900  to  2100;  then  q  =  0.95  and  y  =  0.098.  When  the  super- 
heat was  but  4  degrees  F.,  and  V  =  1770,  q  =  0.915  and  y  =  0.16.  For 
Figs.  126  and  127  no  direct  experiments  were  made,  but  computed  veloci- 
ties from  turbine  tests  indicate  that  for  V  =  200  to  300,  q  =  0.95  and 
y  =  0.1. 

In  all  curved  passages,  centrifugal  force  results,  causing  variation 
in  pressure,  the  maximum  being  at  the  concave  surface.  If  unconfined 
laterally  the  jet  spreads,  reducing  the  efficiency.  Experiments  indicate 
that  the  value  of  q  for  the  passages  between  impulse  blades  for  theo- 


THE  STEAM  TURBINE  221 

retical  steam  velocities  of  from  200  to  2600  ft.  per  sec.,  when  the  jet  is 
unconfined  laterally,  is  given  by  the  formula: 

' 


When  the  jet  is  confined  by  shrouding  as  in  Fig.  130,  the  value  of  q  may  be 
much  increased.  From  (21)  the  velocity  coefficient  for  unshrouded 
guides  and  blades  is  greater  at  high  steam  velocities,  which  is  contrary  to 
the  usual  effect  of  friction.  Peabody  gives  for  flow  through 
guides  and  blades: 

V 

~  10,000 
which  gives: 

(22) 


This  is  no  doubt  intended  for  unworn,  shrouded  blades 
made  by  machining  or  drop  forging. 

Jude  says  that   che    average  maximum  velocity  co- 
efficient for  the  best  form  of  closed-vane  passage  is  about 
0.955,   but  that  the  average  value  of  the  better-known 
turbines  appears  to  be   about  0.92.     However,  various        FIQ   13Q 
phenomena  due  to  the  steam  surrounding  the  vanes  and 
the  transfer  of  steam  from  one  passage  to  another  give  an  equivalent  to 
the  reduction  of  this  value,  sometimes  to  as  low  as  0.6  or  0.7.     Martin 
gives  from  0.68  to  0.72,  while  Rateau  gives  0.75  as  a  fair  average.     Jude 
states  that  he  has  confirmed  Rateau's  results,  but  that  with  special  blades, 
he  has  obtained  values  as  high  as  0.8. 

As  we  have  taken  q  as  the  ratio  of  actual  to  theoretical  velocity  at  the 
exit  of  the  passage,  and  as  the  relative  velocity  in  one  vane  passage  of  an 
impulse  turbine  is  theoretically  constant  up  to  the  entrance  of  the  next 
vane  passage,  the  intervening  space  with  its  losses  may  be  considered  as  a 
part  of  the  passage  and  these  latter  values  of  q  as  applying  to  this  case, 
instead  of  the  higher  values  found  by  experiment  upon  single  rows  of 
blades.  These  higher  values  may  only  be  used  with  other  factors  which 
separately  account  for  other  losses,  a  detailed  study  of  which  may  be 
found  in  the  works  cited.  Jude  says,  however,  that  for  velocicy-stage 
blading  it  is  probably  more  correct  to  take  a  higher  value  of  q,  such  as 
0.92,  as  an  average  in  ascertaining  the  diagram  efficiency.  It  is  possible 
that  as  high  as  0.85  may  be  taken  for  q  in  the  case  of  velocity-stage  tur- 
bines, but  why  it  should  be  greater  than  for  a  simple  impulse  wheel  is  not 
altogether  clear. 


222 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  constants  for  blade  passages  of  the  reaction  turbine  may  best  be 
treated  under  reaction  blading.  A  number  of  practical  nozzle  designs 
are  shown  in  Chap.  XXXIII. 

In  guide  vanes  and  nozzles  formed  by  casting  in  plates  the  pitch 
should  be  as  large  as  is  consistent  with  properly  directing  the  stream. 

Methods  of  drawing. nozzles,  guides  and  blading  are  given  in  Chaps. 
XXXI  and  XXXIII. 

86.  Impulse  Blading. — A  simple  arrangement  of  nozzles  and  blading 
for  an  impulse  turbine  is  shown  in  Fig.  460,  Chap.  XXXIII.  It  will  be 
seen  that  steam  from  the  nozzle  will  sometimes  impinge  upon  several 
blades  at  one  time,  but  in  our  conception  of  the  operation  we  may 
assume  one  pound  acting  upon  a  single  blade. 


FIG.  131. 

The  means  of  determining  the  jet  velocity  has  already  been  given  by 
(14).  The  change  of  velocity  and  the  consequent  change  of  kinetic 
energy  which  the  jet  undergoes  in  its  passage  between  the  blades  may 
best  be  explained  by  the  velocity  diagram,  the  fundamental  form  of  which 
may  be  easily  understood  by  the  crude  apparatus  shown  in  Fig.  131, 
which  needs  little  explanation. 

A  paper  may  be  tacked  to  the  board  A  which  slides  upon  B .  The 
sliding  bar  G  holds  a  pencil  in  its  lower  end,  and  another  pencil  is  held 
by  H  which  is  attached  to  B,  but  made  removable.  C  and  D  may  be 
grooved  pulleys  fastened  together,  the  ratio  of  diameters  being  the  ratio 
of  jet  to  blade  velocity. 

To  draw  a  diagram,  disconnect  the  cord  E  and  remove  H]  then  turn 
the  pulleys  so  as  to  slide  bar  G  and  draw  line  1-2.  This  represents  the 
direction  and  velocity  of  the  jet  from  the  nozzle.  Connect  cord  E  and 


THE  STEAM  TURBINE 


223 


disconnect  cord  F  after  replacing  G  to  its  original  position  and  removing 
the  pencil.  Place  H  so  that  its  pencil  point  is  at  2  and  turn  the  wheel 
through  the  same  angle  as  before,  drawing  line  2-3.  This  represents  the 
direction  of  motion  and  velocity  of  the  turbine  blade.  Move  A  back  so 
the  pencil  in  H  is  at  2  again,  than  remove  H,  connect  cords  E  and  F, 
place  pencil  of  G  on  paper  at  1  and  turn  wheel  through  same  angle  as  the 
two  previous  times.  The  relation  of  the  movements  of  G  and  A  will  be 
that  of  the  jet  to  the  moving  blades,  and  the  line  1-3  will  be  drawn  on 
the  paper.  This  is  the  direction  of  the  pencil  in  G  with  reference  to  the 
paper,  or,  it  represents  the  direction  of  the  jet  and  its  velocity  in  relation 
to  the  moving  blade  at  the  time  the  steam  begins  to  enter  the  blade 
passage,  and  is  known  as  relative  velocity.  If  friction  were  absent  and 
there  were  no  other  forces  tending  to  change  the  velocity  (such  as  expan- 
sion in  the  blades  of  the  reaction  turbine),  the  relative  velocity  would  be 
constant,  but  its  direction  would  change  due  to  the  influence  of  the  blade 
passage  which  now  controls  its  flow. 

The  lines  1-2  and  2-3  represent  velocities  with  reference  to  a  fixed 
object  upon  the  earth's  surface  and  are  known  as  absolute  velocities.  As 
in  marine  service  the  turbine  is  in  motion,  the  fixed  parts  of  the  turbine, 
such  as  the  casing,  may  be  considered  the  zero  of  motion  from  which 
absolute  velocity  is  measured.  By  changing  the  angle  of  the  rod  G, 
or  the  relative  diameters  of  C  and  Z>,  velocity  diagrams  of  different  form 
may  be  drawn;  it  is  simpler,  however,  to 
abandon  the  apparatus,  which  was  only 
mentioned  with  the  thought  of  making 
the  matter  plain  to  some  who  might  not 
easily  grasp  the  principle. 

A  simple  velocity  diagram,  including 
the  diagram  for  exit  from  the  blade,  is 
shown  in  Fig.  132.  The  concave  surface 
of  the  blade  against  which  the  driving 
force  is  exerted  is  shown  by  the  heavy 
line,  the  back  of  the  blade  being  omitted. 
The  blade  is  shown  tangent  to  VN,  the 
relative  velocity  at  entrance,  so  that  the 
jet  may  enter  without  shock;  this  is  the 

usual  treatment,  but  more  will  be  said  upon  the  subject  presently.  The 
blade  curve  must  be  tangent  to  line  Vx  for  impulse  blades,  to  insure  the 
direction  of  flow. 

The  blade  shown  in  Fig.  132  is  symmetrical,  angles  0  and  <f>  being 
equal;  and  as  friction  is  neglected,  Vx,  the  relative  velocity  at  exit,  is 


\ 


\ 


FIG.  132. 


224 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


equal  to  VN.    S  is  the  velocity  of  the  blade  (at  the  pitch  circle)  and  must 
be  equal  in  both  entrance  and  exit  diagrams. 

If  in  Fig.  131  bar  G  were  changed  to  the  angle  5  of  Fig.  132  and  the 
line  VR  drawn  with  the  paper  stationary  instead  of  the  line  1-2  of  Fig.  131, 
the  movement  of  both  bar  and  paper  would  draw  the  line  Vx  for  the  line 
1-3,  which  must  then  be  relative  velocity  as  already  stated.  This  shows 
that  line  VR  of  Fig.  132  represents  absolute  velocity,  and  it  is  known  as 
residual  velocity.  The  reduction  of  the  velocity  of  the  steam  in  its  pas- 
sage through  the  blades  is  the  difference  between  V  and  VR,  both  being 

absolute  velocities.  The  kinetic 
energy  due  to  residual  velocicy  VR 
is  the  energy  lost,  and  should  be  kept 
as  small  as  practicable. 

The  absolute  entrance  velocity  V 
of  Fig.  132  may  be  resolved  into  two 
components,  one  parallel  to  the  shaft 
center  and  represented  by  line  1-2, 


133 


called  the  velocity  of  flow;  the  other 
in  the  direction  of  blade  motion  by 
line  2—3,  called  the  velocity  of  whirl. 
Likewise  the  absolute  exit  velocity, 
or  residual  velocity  may  be  re- 
solved into  4-6,  the  exit  velocity  of 

flow,  and  5-6,  the  exit  velocity  of  whirl.  The  latter,  in  the  diagram  given 
is  negative,  being  opposite  in  direction  to  the  blade  movement ;  but  the 
momentum  due  to  it  drives  the  blade  forward  so  is  considered  positive. 
A  more  general  treatment  may  be  given  by  considering  friction  and 
assuming  an  unsymmetrical  blade;  such  a  diagram  is  Fig.  133,  to  which 
the  same  general  definitions  apply. 

The  kinetic  energy  supplied  to  the  blade  is  due  to  the  jet  from  the 
nozzle  at  velocity  V,  and  is,  in  B.t.u.  for  1  Ib. ; 

V2  V 

A^g  =  778  X  64.32 


V223.' 


The  rejected  energy  due  to  residual  velocity  is: 


/  V« 
\223.77 


and  the  energy  lost  by  friction  is: 


/-- 

\223.77 


223.7 


THE  STEAM  TURBINE  225 

or: 


The  efficiency  of  the  blades,  or  diagram  efficiency  eD,  is  then  given  by: 

^^^'••^_v'-v*-jM-*>  ;        (23) 

The  force  of  impact  is  the  product  of  mass  and  velocity  change  per 
second.  It  is  obvious  that  the  change  of  tangential  velocity  of  the  steam 
produced  by  its  impingement  upon  the  turbine  blade,  is  the  difference  be- 
tween the  tangential  velocity,  or  velocity  of  whirl,  at  entrance  and  exit; 
which  is: 

V  cos  a  —  VR  cos  d. 

The  direction  of  blade  motion  having  the  plus  sign,  the  value  of  VR  cos  d 
is  negative.  To  obviate  the  necessity  of  considering  which  sign  this 
value  has,  we  may,  in  view  of  the  fact  that  the  tangential  component  of 
Vx  is  always  negative,  and  S  always  positive,  write: 

VR  cos  d  =  —  Vx  cos  0  H-  S 
Then  the  tangential  velocity  change  is: 

V  cos  a  -  (  -  Vx  cos  0  +  S) 
or: 

V  cos  a  +  Vx  cos  0  —  S 

Then,  substituting  qVN  for  Vx,  the  tangential  force  due  the  flow  of 
w  Ib.  of  steam  per  sec.  is: 

fw  =  -(V  cos  a  +  qVN  cos  </>  -  S)  =  -  Vw  .(24) 

y  y 

This  is  the  load  on  all  blades  receiving  the  steam;  to  find  the  load  per 
blade  divided  by  the  number  of  blades. 

Similarly,  the  axial  pressure  on  the  blades  is  due  to  the  difference  of 
the  velocity  of  flow  at  entrance  and  exit,  and  is  given  by  the  expression: 

fp  =  ™(V  sin  a  -  qVN  sin  0)  =  -  Vf  (25) 

*/  *7 

This  is  known  as  axial  thrust  and  must  be  provided  for  by  a  thrust 
bearing.  If  fF  is  negative  the  thrust  is  in  the  opposite  direction  from  the 
velocity  of  flow.  Formulas  (24)  and  (25)  are  general  and  apply  to  reac- 
tion blades  also. 

15 


226 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


A  graphical  construction  for  the  parenthetical  quantities  of  (24)  and 
(25)  is  given  in  Fig.  134,  which  may  be  drawn  to  scale  and  then  measured 
as  an  alternative.  Fig.  134  is  a  reproduction  of  Fig.  132  with  the  line 
Vx  (=  qVir)  produced  on  the  upper  side  of  the  center  line  of  blades. 

The  useful  work  per  second  done  upon  the  turbine  wheel  isfwS  ft.  lb.; 
and  as  wV2/2g  is  the  ft.  lb.  of  energy  supplied  by  the  nozzles  per  second, 
the  diagram  efficiency  is: 


eD 


~  (V  cos  a  +  qV  cos  <t>  -  S) 


(26) 


The  quantity  in  parenthesis  may  be  calculated,  or  measured  from  the 
diagram,  Fig.  134.  Formulas  (23)  and  (26)  give  the  same  result,  the 
latter  being  more  convenient. 


FIG.  134. 

Formula  (26)  may  also  be  written: 


eD 


From  Fig.  133: 


2  y  (cos  a  +  q  -^  cos  <f>  -y 
VN  cos  6  =  V  cos  a  —  S 

s 


v         cose 

Substituting  in  Formula  for  eD  and  rearranging  gives: 

S  S 


Also  VN  sin  6  =  V  sin  a 


THE  STEAM  TURBINE 


227 


or, 


\VN  _  sin  a 
~V    ~  sin  e 
Equating  with  the  value  of  VN/V  just  given: 


tan  6 


sin  a 


S 

COS  a  —  =. 


(28) 


When  S/  V  and  a  are  known  K  may  be  found,  and  cos  0  may  be  deter- 
mined from  (28).  Table  38  gives  values  of  K  and  cos  0  for  different 
values  of  a  and  S/V;  then  by  assuming  different  values  of  q  and  <f>,  eD  may 
be  quickly  determined.  Within  reasonable  limits  it  is  apparent  that 
decreasing  0  increases  the  efficiency;  and  that  for  a  given  value  of  cos 
0/cos  0  and  q  the  efficiency  varies  directly  as  K.  Then  it  may  be  seen 
that  decreasing  the  angle  a  increases  the  efficiency,  and  for  maximum 
efficiency  such  a  value  of  S/V  must  be  chosen  which  will  make  K  a  maxi- 
mum. This  may  be  found  by  equating  the  first  derivation  of 


S  S2 

^-T  COS  a  —  v^. 


to  zero,  which  gives : 


cos  a 


(29) 


TABLE  38 


a 

S/V 

0.3 

0.4 

0.5 

0.6 

K 

0 

cos  & 

K 

I 

cos  0 

K 

0 

cos  0 

K 

e 

cos  e 

15 
20 
25 

0.400 
0.383 
0.364 

21°-13' 
26°-53' 
34°-58' 

0.932 
0.892 
0.819 

0.453 
0.432 
0.405 

24°-37' 
32°-25' 
39°-54' 

0.929 
0.844 
0.767 

0.466 
0.439 
0.406 

28°-54' 
37°-55' 
46°-8/ 

0.875 
0.789 
0.693 

0.440 
0.407 
0.366 

35°-ll' 
45°-51' 
54°-5' 

0.817 
0.696 
0.586 

This  will  give  the  maximum  value  of  K  for  any  given  value  of  a.  Then 
S  is  one-half  the  velocity  of  whirl,  and  neglecting  friction,  Formula 
(27)  for  symmetrical  blades  reduces  to: 

eD  =  cos2  a 
or  with  unsymmetrical  blades: 


eD  = 


COS' 


which  may  give  a  greater  efficiency  and  tend  to  reduce  the  effect  of  fric- 
tion. 


228 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


87.  Practical  Notes  on  Impulse  Blades. — The  velocity  coefficiencs 
for  impulse  blades  and  guides  are  given  at  the  latter  end  of  the  preceding 
paragraph;  these  are  for  shrouded  blades,  which  are  practically  always 
used  with  impulse  turbines. 

While  the  nozzle  exit  is  reduced  slightly  from  the  area  theoretically 
indicated,  as  stated  in  Par.  84,  the  blade  and  guide  passages  must  not  be 
restricted  as  this  may  lead  to  throttling.  It  is  probably  well  to  consider 
all  effects  of  friction  and  use  Formula  (17)  or  (19)  in  determining  exit 
areas.  As  the  clearance  space  between  nozzles  and  blades  probably  gives 
rise  to  friction  loss  with  a  possible  slight  spreading  of  the  jet,  the  radial 
length  of  the  blades  may  be  slightly  greater  than  the  radial  width  of 
the  nozzles;  possibly  if  this  were  made  equal  to  what  the  nozzle 
width  would  be  by  making  allowance  for  the  drying  effect  of  friction, 
the  condition  would  be  properly  met.  Some  designers  increase  the 
blade  length  beyond  this  rather  arbitrarily  to  avoid  any  possibility  of 
choking  (see  also  velocity-stage  blading);  with  round  nozzles,  how- 
ever, the  blade  areas  would  exceed  nozzle  areas  and  choking  would  not 
be  likely. 

As  stated  earlier  in  this  paragraph,  the  entrance  angle  of  the  blade  is 
usually  assumed  equal  to  the  angle  6  at  which  the  steam  enters  the  blade; 
the  idea  is  that  with  such  a  construction  the  steam  slides  into  the  blade 
passage  without  shock,  and  that  friction  is  thus  reduced  to  a  minimum . 
According  to  Martin,  this  is  an  error,  and  better  practical  results  have 

been  obtained  by  making  the  blade  angle 
some  5  to  15  degrees  greater  than  the  angle  6, 
Fig.  133.  This  is  shown  in  Fig.  135.  If 
this  is  not  done,  the  decreasing  relative 
velocity  due  to  friction  changes  the  direc- 
tion in  which  the  steam  attempts  to  flow, 
exerting  pressure  on  the  blade  surface  form- 
ing the  back  of  the  passage  and  tends  to 
retard  the  motion  of  the  wheel.  The  tra- 
jectory diagrams,  Fig.  150,  are  drawn  with 
the  entrance  angle  of  the  moving  blades 
greater  than  6,  and  it  may  be  seen  that 
the  stream  from  the  nozzle  is  deflected 
backward  slightly  upon  joining  the  tra- 
jectory curves,  showing  a  forward  impulse.  Had  the  entrance  angle 
been  equal  to  0,  the  opposite  would  have  been  true,  as  friction  is  taken 
into  account  in  these  diagrams.  This  is  explained  in  Par,  92. 

The  advisability  of  making  .<f>  less  than  0  must  be  left  to  the  designer's 


FIG.  135. 


THE  STEAM  TURBINE 


229 


judgment;  it  is  usually  done  and  no  doubt  increases  the  efficiency  if  not 
carried  too  far. 

Jude  says  that  the  drying  of  the  steam  due  to  blade  friction  is 
improbable,  as  any  water  present  is  thrown  to  the  outside  of  the  curved 
path  by  centrifugal  force  and  is  difficult  to  reevaporate. 

No  definite  rule  controls  the  blade  length.  For  reaction  turbines 
Martin  gives  a  minimum  length  of  ^5  the  drum  diameter  for  stationary 
turbines  and  1/75  for  marine  turbines,  the  latter,  however  involving  a  loss 
of  efficiency.  The  maximum  length  for  the  low-pressure  end  is  given 
as  1/5  the  pitch  diameter.  These  ratios  m  may  be  used  as  a  guide  and 
should  be  as  reliable  for  impulse  turbines. 

There  seems  to  be  no  rational  method  of  determining  the  pitch  of 
impulse  blading,  which  in  practice  ranges  from  48  to  68  per  cent,  of  the 
axial  width.  It  would  seem  reasonable  to  make  the  pitch  as  large  as 
possible  and  still  properly  direct  the  steam,  thus  reducing  the  friction 
surface,  and  it  is  probable  that  the  greater  the  radius  of  curvature  of  the 
blade,  the  greater  the  pitch  may  be.  A  method  of  drawing  blades  is  given 
in  Chap.  XXXI,  by  which  the  blades  of  Figs.  149  and  150  were  drawn. 

The  speed  at  the  pitch  circle  of  simple  impulse  turbines  such  as  the  De- 
Laval  ranges  from  500  to  1400  ft.  per  second.     For  pressure-stage  impulse 
turbines  a  range  from  300  to  650  ft.  is  used,  some  large  modern  turbines 
having  a  blade-tip  velocity  as  high  as  950 
ft.  per  sec. ;  in  this  case  the  blade  length  of 
the    last    wheel    exceeds    J4  of  the  pitch 
diameter,  or  m  is  greater  than  J^.    The 
higher  velocities  necessitate  great  care  in 
the  designing  of  wheels  and  shafts  which 
is  discussed  in  Chaps.  XXXI  and  XXXII. 

88.  Velocity-stage  Impulse  Turbine. — 
For  the  purpose  of  studying  the  principle 
of  speed  reduction  of  the  velocity-stage 
turbine,  frictionless,  symmetrical  blades 
and  guides  will  first  be  considered.  A  dia- 
gram for  a  single  stage  with  zero  exit 
velocity  of  whirl  is  given  in  Fig.  136.  In 
this  diagram,  6  =  $,  and  it  is  obvious 
that  the  triangle  formed  by  the  exit 

diagram  is  equal  to  that  formed  by  the  dotted  lines  at  the  left  of  the  en- 
trance diagram.  These  are  shown  together  in  Fig.  137.  Retaining  the 
zero  exit  velocity  of  whirl,  assume  that  the  blade  speed  S  is  to  be  made 
one-half  that  shown  in  Fig.  137.  This  is  done  by  dividing  the  base  line 


FIG.  136. 


230 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


into  two  parts,  forming  four  triangles  as  shown  in  Fig.  138.  Disposing 
these  triangles  in  a  manner  similar  to  Fig.  136  gives  Fig.  139,  which  is  a 
double  diagram,  having  two  rows  of  moving  blades  and  one  row  of  guides, 
which,  except  that  there  is  no  expansion  in  them,  serve  as  nozzles  for  the 


FIG.  137. 


FIG.  138. 


Subscripts  1  and  2  are  for  the  first  and  second 


blades  of  the  second  wheel. 

stages  respectively. 

From  Fig.  139  it  may  be  seen  that  for  the  assumption  made,  the  blade 

speed  varies  inversely  as  the  number  of  velocity  stages  operated  from  one 

set  of  nozzles.     In  practice  there  is  the  ever-present  friction,  and  the 

blades  and  guides  are  not  always  symmetrical, 
especially  the  latter;  as  when  angle  «2  is  re- 
duced— sometimes  to  equal  6,  in  2-stage  wheels 
— the  blades  are  not  so  flat  in  form.  Fig.  140 
is  a  diagram  for  a  2-stage  wheel  with  these 
modifications. 


FIG.  140. 


Three  or  more  rows  of  moving  blades  may  be  treated  in  the  same 
manner,  the  ratio  S/V  being  reduced  correspondingly.  The  tangential 
impulse  on  the  blades  may  be  found  for  each  moving  row  by  (24),  or 


THE  STEAM  TURBINE  231 

graphically  from  Fig.  134,  using  subscripts  1,  2,  3,  etc.,  for  the  successive 
stages.     The  total  energy  is  given  by: 

S  (fwi  +  fw2  +  fws,  etc.) 

Letting  Vw  =  V  cos  a  +  qVN  cos  <f>  —  S,  the  parenthetical  quantity 
of  (24),  the  total  energy  is: 


and  as  the  energy  of  the  jet  from  the  nozzles  is  wVi2/2g,  the  diagram  effi- 
ciency is  : 

rtCf 

*°  =        (Wi  +  v™  +  v"*>  etc-) 


From  the  equations  of  paragraph  86  it  would  seem  that  whatever  the 
velocity  coefficients  were,  maximum  efficiency  would  be  expected  when 
the  diagram  is  constructed  so  that  : 

Q  _  V  cos  a 
~2~ 

for  the  last  stage,  if  this  does  not  necessitate  too  small  a  value  of  a. 

The  tangential  pressure  on  the  blades  may  be  found  from  (24),  and 
this  gives  the  proportion  of  work  done  by  each  set  of  moving  blades.  The 
axial  thrust  is  the  sum  of  the  thrust  on  all  sets  of  moving  blades,  and  may 
be  found  from  (25),  using  the  proper  subscripts. 

In  Fig.  139  it  may  be  seen  by  inspection  that  V\  cos  «i  =  4S,  and 
VNi  cos  <£i  =  3/S.  Then  VWi  =  6£;  also  V2  cos  «2  =  2S,  VN2  cos  fa  =  S. 
Then  Vw*  =  2S,  or: 

fwi  =  3/Va 

In  like  manner  it  may  be  shown  that  for  symmetrical  blades  and  guides, 
neglecting  friction,  the  ratio  of  impulse,  beginning  with  the  first  row  is  : 

For  3  stages  ....................  5,  3,  1 

For  4  stages  ....................  7,  5,  3,  1 

For  5  stages  ....................  9,  7,  5,  3,  1 

This  is  also  the  proportion  of  work  done  by  the  respective  stages.  These 
proportions  vary  with  actual  blades,  but  give  some  idea  of  the  value  of  the 
lower  stages. 

89.  Practical  Notes  on  Velocity-stage  Turbines.  —  The  velocity 
coefficients  for  velocity-stage  turbines  vary  considerable,  making  the 
diagram  efficiency  calculated  from  (30)  rather  uncertain.  With  the 


232  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

low  value  of  q  sometimes  found  by  experiment,  but  little  work  would  be 
done  by  the  lower  stages,  with  even  the  possibility  that  the  extra  work 
of  dragging  one  or  more  idle  wheels  be  imposed  upon  the  turbine,  espe- 
cially at  light  loads  with  the  steam  throttled.  It  is  therefore  seldom 
that  more  than  three  velocity  stages  are  employed  in  stationary  turbines, 
although  four  are  sometimes  used  for  the  high-pressure  stage  (see  Par. 
91)  of  marine  turbines,  and  as  many  as  five  for  the  high-pressure  stage  of 
the  reversing  turbine,  in  which  economy  of  space  is  more  important  than 
economy  of  steam  for  the  relatively  short  periods  it  is  in  use. 

It  is  evidently  important  to  consider  all  points  which  tend  toward 
efficiency.  The  radial  length  at  entrance  of  blades  and  guides  may  be 
slightly  greater  than  the  exit  length  of  the  preceding  guides,  blades  or 
nozzles,  to  avoid  choking;  and  the  ratio  of  exit  to  entrance  length  of  each 
row  of  blades  and  guides  may  be  found  from  (17)  or  (19). 

In  some  turbines,  even  though  the  blades  are  not  symmetrical,  the 
blade  length  is  the  same  at  entrance  and  exit,  probably  for  constructional 
reasons.  Martin,  commenting  upon  a  Curtis  marine  turbine  with  four 
velocity  stages  in  the  first  pressure  stage  says:  "The  bucket  angle  of  the 
first  row  at  entrance  is,  it  will  be  seen,  28  degrees,  while  at  discharge  it 
is  22  degrees.  From  this  it  follows  that  the  space  between  the  buckets 
is  narrower  at  discharge  than  at  entrance,  but  this  is  in  accord  with  the 
fact  that  the  stream  of  fluid  as  it  passes  through  the  bucket  tends  to 
spread  laterally,  and  consequently  becomes  thinner.  It  will  further  be 
observed  that  each  successive  bucket  is  longer  than  its  predecessor.  This 
is  necessary,  because  in  each  bucket  the  fluid,  besides  thinning,  as  already 
mentioned,  also  loses  velocity,  and  thus  a  greater  steam  way  is  required  at 
each  successive  set  of  buckets." 

As  the  turbine  referred  to  had  shrouded  blades,  the  spreading  and 
thinning  of  the  stream  could  occur  only  if  the  passages  did  not  run  full 
at  entrance.  Concerning  the  necessary  increase  in  area  due  to  decreasing 
velocity,  this  is  provided  for  by  the  increasing  angles ;  the  areas  are  propor- 
tional to  the  sines  of  the  angles,  and  between  two  rows  of  blades  the  prod- 
uct of  the  sine  of  the  angle  and  the  velocity  is  the  same,  so  that  neglecting 
friction  in  the  space  between  the  rows,  and  spilling,  the  exit  length  of 
one  row  and  entrance  length  of  the  next  could  be  the  same.  Had  the 
blades  in  question  been  computed  by  (19),  with  an  increase  of  entrance 
over  the  exit  of  the  preceding  row  as  already  mentioned,  then  if  in  each 
case  both  blade  edges  had  been  made  the  same  length  as  the  longer,  a 
similar  result  would  have  been  obtained;  any  difference  would  probably 
be  due  to  the  arbitrary  increase  of  entrance  allowed. 

In  some   cases,   after  the  exit   length  of  the  last  blade  has  been 


THE  STEAM  TURBINE  233 

found,  straight  lines  from  its  extremities,  drawn  to  the  extremes  of 
nozzle  opening,  give  rather  arbitrarily  the  dimensions  of  all  blades  and 
guides. 

Should  the  thickness  factor  of  the  blades  not  be  the  same,  allowance 
may  be  made,  but  this  is  usually  negligible. 

In  Par.  97,  Fig.  163  shows  the  arrangement  of  blades  for  two  velocity 
stages  of  the  first  pressure-stage  of  the  turbine  designed  as  an  example  of 
formula  application.  Formula  (19)  was  used  to  determine  blade  lengths, 
neglecting  the  drying  due  to  friction,  and  the  increase  of  entrance  height 
over  the  preceding  exit  height;  then  the  form  was  modified  as  explained. 
The  ratio  of  maximum  height  of  the  last  blade  to  the  nozzle  diameter  is 
1.71. 

A  similar  layout  for  a  Curtis  turbine  shown  by  Martin  gives  a  ratio 
of  1.73;  another,  however,  in  which  each  row  has  a  constant  height,  has 
a  ratio  of  but  1.2. 

It  seems  probable  that  Formula  (19)  is  practical,  with  modifications 
suggested  and  illustrated  in  Figs.  164  and  165.  The  drying  effect  may  be 
taken  into  account  if  desired,  although  such  drying  is  discredited  by 
Jude,  as  previously  stated. 

In  spacing  blades,  the  pitch  is  sometimes  made  the  same  for  all  rows, 
but  is  often  increased  as  the  entrance  and  exit  angles  are  increased,  which 
seems  reasonable.  Methods  of  laying  out  blades  are  given  in  Chap. 
XXXI. 

Axial  clearance  between  nozzles  and  blades  has  less  influence  upon 
economy  than  clearance  between  blades  and  guides,  and  experiment  has 
shown  that  the  loss  due  to  the  latter  increases  rapidly  with  increase  of 
clearance.  Axial  clearance  should  therefore  be  kept  as  small  as  reliability 
of  operation  will  permit. 

As  stated  in  Par.  86,  the  reduction  of  exit  angle  of  blades  is  a  matter 
of  judgment  but  is  customary,  and  the  exit  angle  of  guides  should  be  re- 
produced in  order  that  the  following  blades  may  not  be  too  flat:  a  com- 
parison of  Figs.  139  and  140  will  make  this  clear.  Prof.  Peabody  says 
that  a  conservative  and  convenient  arrangement  is  to  make  the  exit 
guide  angle  equal  to  the  preceding  blade  angle;  or,  az  =  0i,  0:3  =  #2,  etc. 

The  entrance  angles  of  all  moving  blades  should,  according  to  Martin, 
be  made  from  5  to  10  degrees  greater  than  6,  as  stated  in  Par  86. 

Values  of  eD  found  from  practice  are  given  by  Martin  as  follows: 

2-stage eD  =  0.72 

3-stage eD  =  0.65 

4-stage eD  =  0.52 


234  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

He  states  that  for  maximum  efficiency  the  ratio  of  V/S  for  a  4-stage  tur- 
bine should  be  about  11.  These  values  may  be  used  as  a  check.  Velocity 
coefficients  are  given  in  Par.  84. 

90.  Pressure-stage  Impulse  Turbine.  —  The  equation  connecting 
flow  with  heat  drop  is  given  by  (14),  Par.  84,  and  is: 

V  =  223.7V(1  -  y)  (Cl  -  C2) 

Taking  Ci  —  €2  as  the  total  adiabatic  heat  drop  from  steam  pipe  to  con- 
denser, we  may  divide  it  into  a  number  of  parts  as  explained  in  Par.  10, 
Chap.  IV.  This  will  cause  a  series  of  pressure  drops  until  the  exhaust 
pressure  is  reached,  the  pressure  in  the  successive  compartments  of  the 
turbine  being  maintained  by  properly  proportioning  the  nozzle  areas  con- 
necting them,  so  that  the  inflow  and  outflow  for  each  compartment  in  a 
given  time  is  the  same,  and  equilibrium  is  not  secured  until  a  certain 
definite  piressure,  predetermined  for  a  given  weight  of  steam  per  second, 
is  attained  in  each  stage. 

For  simplicity,  let  nP  be  the  number  of  equal  pressure  stages;  that  is, 
the  heat  drops  being  equal.  We  then  have  for  the  velocity  of  the  jets 
flowing  into  each  stage,  assuming  y  to  be  the  same  in  each  stage  and 
neglecting  certain  influences  to  be  mentioned  later: 


V  =  223.7     (l  -2/)--  (31) 

\  tip 

If  the  pitch  circle  of  the  wheels  is  the  same  in  all  stages,  and  the  nozzle 
and  blade  angles  the  same,  the  velocity  diagram  will  be  the  same  for 
each  stage,  and  from  (31)  it  may  be  seen  that  the  blade  velocity  is  in  in- 
verse proportion  to  the  square  root  of  the  number  of  pressure  stages. 

In  the  nozzle  example  of  Par.  84  for  a  noncondensing  turbine,  the 
velocity  V  from  the  nozzle  is  2750.  Assume  Fig.  133  as  the  velocity 
diagram,  in  which  S/V  is  %.  Then  'S,  the  blade  velocity  is  2750/3,  or 
916  ft.  per  sec.  Now  assume  the  same  total  heat  drop  for  a  turbine 
with  four  pressure  stages.  The  blade  speed  is  then: 


If  the  wheels  of  the  two  turbines  are  the  same  diameter,  the  r.p.m. 
of  the  4-stage  turbine  is  one-half  that  of  the  single  stage.  Pressure- 
stage  turbines  may  then  be  used  when  lower  speeds  are  desired.  The 
velocity  diagram  may  be  drawn  as  for  -a  simple  turbine  and  used  for  a 
pressure-stage  turbine  with  any  number  of  stages  by  changing  the  scale. 

Intermediate  pressures  may  be  found  by  means  of  entropy  table  or 
chart.  Taking  the  same  4-stage  turbine  but  neglecting  friction  and 


THE  STEAM  TURBINE  235 

assuming  adiabatic  expansion,  if  we  find  the  heat  content  C  for  each  stage, 
the  corresponding  pressure  may  be  read  from  table  or  chart. 

During  expansion  in  the  first  set  of  nozzles,  a  quantity  of  heat  equal 
to  (Ci  —  C2)/nP  was  used,  leaving  the  heat  content  in  the  first  stage: 

A  ~      l  "         nP 
in  the  second  stage: 

"l—  ft  „  «Cl-C, 


_     /"Y        Q 


nP  nP 

in  the  third  stage: 


nP  nP 

and  in  the  fourth  stage: 


This  may  be  applied  to  any  number  of  stages.  Proceeding  with  the 
example:  At  164.8  Ib.  absolute  and  entropy  1.56,  Ci  =  1193.3.  At 
14.7  Ib.  absolute  and  the  same  entropy,  C2  =  1018.  Then  as  (Ci  — 
C,)/4  =  43.825: 

First  stage  ................  CA  =  1193.3  -    43.825  =  1149.475,  and  P  =  95.715 

Second  stage  ...............  CB  =  1193.3  -    87.650  =  1105.650,  and  P  =  53.568 

Third  stage  ...............   Cc=  1193.3  -  131.475  =  1061.825,  and  P  =  28.758 

Fourth  stage  ...............  CD  =  C2  =  1018  and  P  =  14.700 

Pressures  were  found  by  interpolation,  using  Peabody's  entropy 
table.  The  heat  quantity  43.825  B.t.u.  is  then  available  in  each  stage, 
and  may  be  illustrated  by  the  entropy  diagram,  Fig.  141,  which  is 
divided  into  four  equal  parts  by  the  pressures  given.  'To  make  the 
problem  a  little  plainer,  a  diagram  of  a  4-stage  turbine  is  given  in  Fig. 
142,  showing  the  pressure  distribution. 

Effect  of  Heal  Factor.  —  The  example  and  Fig.  141  assume  a  perfect 
turbine,  in  which  the  heat  factor  (see  Par.  41,  Chap.  VIII)  is  unity.  Due 
to  the  diagram  efficiency  being  less  than  unity,  to  disc  friction  and  other 
disturbing  influences,  the  heat  factor  is  less  than  unity,  and  a  part  of  the 
kinetic  energy  of  the  jet  is  converted  into  heat,  increasing  the  dry  ness 
factor,  volume  and  entropy.  This  change  takes  place  during  the  entire 
pressure  drop,  and  while  we  do  not  know  the  exact  form  of  the  curve, 
we  know  the  overall  heat  factor  from  tests  and  may  thus  find  the  entropy 
of  exhaust  pressure,  and  assuming  the  same  percentage  of  loss  in  each 
stage,  may  fix  several  points. 

Assume  the  total  heat  factor  F  to  be  0.6;  then,  neglecting  radiation  loss 
0.4  X  175.3  =  70.12  B.t.u.  are  returned  to  the  steam  at  exhaust  pressure 


236 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


as  heat,  making  the  heat  content  at  14.7  lb.,  1088.12  B.t.u.     Interpolating 
between  entropies  1.66  and  1.67; 


1091.9 
1085.1 


1088.12 
1085 . 10 


1.67 
1.66 


1.6600 
0.0044 


6 . 8  is  to        3 . 02  as  0 . 01  is  to  0 . 0044.     Entropy  is  1 . 6644 


32°F 


P=/64.S 


FIG.  141. 


Interpolating  between  entropies  is  usually 
unnecessary  for  practical  work,  especially 
with  few  pressure  stages.  Even  with  the 
nearest  entropy  1.66,  the  increase  of 
entropy  is  0.1  between  164.8  and  14.7  lb. 
Fig.  143  shows  the  increase  of  entropy, 
the  dotted  lines  being  the  assumed  curves 
of  entropy  change  due  to  nozzle  friction, 
and  the  full  curves  at  the  right  the  gen- 
eral curve  of  entropy  increase.  The  dis- 
tance from  the  lower  end  of  the  dotted 
curves  to  the  full  curve  represents  the 
increase  of  entropy  due  to  the  change  of 


////////////////////////A         Y/ 

=3=1 

28.  7S8 


Ht=? 


i '4.7 


FIG.  142. 


the  kinetic  energy  of  residual  velocity  into  heat  at  constant  pressure,  and 
this  is  the  larger  part  of  the  change.  The  diagram  is  exaggerated  for 
illustration. 

The  shaded  areas  represent  the  available  adiabatic  heat  drop  for  each 
stage,  and  it  is  obvious  that  if  these  areas  are  to  be  equal,  the  interme- 
diate pressure  must  be  lower  than  for  Fig.  141.  It  must  be  remembered 


THE  STEAM  TURBINE 


237 


that  the  work  done  is  no  more  than  before,  but  assuming  an  equal  division, 
the  work  per  stage  is: 

(32) 


32° F 


HP 

This  is  a  larger  fraction  of  one  shaded  area  of  Fig.  141  than  of  one  area  of 

Fig.  143,  the  former  being  equal  to  (Ci  —  C2)/nP,  and  the  fraction  the  total 

heat  factor/'7;  for  the  latter,  the  area 

being  greater  than  (Ci  —  C2)/Wp,  the 

fraction  is  less  than  F  and  is  known 

as  the  heat  factor  per  stage,  Fs.     This 

is    also    sometimes    known    as   the 

hydraulic   efficiency,    and   the   ratio 

F/FS  as  the  reheat  factor,  FR. 

For  a  single-stage  turbine  F 
equals  Fs.  The  fact  that  F/FS  is 
greater  than  unity  for  two  stages 
and  increases  (slowly)  as  the  num- 
ber of  stages  increases  must  not 
lead  to  the  conclusion  that  increas- 
ing the  number  of  stages  increases 
the  efficiency.  The  over-all  heat 
factor  is  based  upon  ideal  perform- 
ance, viz.,  adiabatic  expansion  from 
the  maximum  to  the  minimum 
temperature,  of  the  heat  available 
at  maximum  temperature,  and  not 
upon  an  uncertain  lot  of  rejuve- 
nated heat  collected  along  the 
downward  path.  The  increased 
available  heat  of  the  multi-stage 
turbine  is  produced  by  the  failure 
of  the  preceding  stages  to  utilize  it, 
being  degraded  by  friction  and  re- 
ceived again  at  lower  temperatures, 
and  its  use  must  be  accompanied  by 
decreased  efficiency. 

That  Fs  is  less  than  F  does  not  seem  inconsistent;  it  assumes  that  the 
ratio  of  the  losses  attending  the  operation  of  one  turbine  wheel  to  the  work 
done  in  one  stage  of  a  multi-stage  turbine,  is  greater  than  the  ratio  of 
loss  due  to  the  wheel  of  a  single-stage  turbine  of  equal  power  to  the  entire 
work  of  the  turbine.  This  could  be  true  if  the  wheel  loss  of  the  simple 


FIG.  143. 


238 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


turbine  were  any  less  than  the  sum  of  the  losses  of  the  wheels  of  the  multi- 
stage turbine,  which  may  reasonably  be  assumed  to  be  the  case. 

Any  actual  superiority  of  multi-stage  over  simple  turbines  is  due  to 
size  of  unit  and  other  features  accompanying  the  design  of  the  larger 
units. 

The  determination  of  Fa  by  calculation,  from  the  results  of  laboratory 
experiments  to  determine  the  value  of  the  factors  upon  which  it  depends, 
is  not  as  reliable  as  the  determination  of  F,  the  over-all  heat  factor,  by 
turbine  tests.  If  Fs  were  known,  we  could,  by  trial  and  error,  with  the  aid 
of  the  intermediate  pressures  found  by  the  assumption  of  constant 
entropy,  find  the  actual  intermediate  pressures  which  would  give  equal 
work,  or  for  any  division  of  work.  Or  we  may  assume  Fs  and  check  our 
results  by  comparison  with  F,  which  must  after  all  be  only  an  assumed 
value. 

Peabody's  Direct  Method. — The  simplest  method  of  determining  the 
reheat  factor  FR  for  saturated  steam,  and  of  applying  it  to  determine 
intermediate  pressures  for  any  heat  distribution  is  given  by  Prof. 
Peabody  in  The  Steam  Turbine,  and  it  will  be  given  here  with  some 
modifications. 

The  entropy  table,  T<f>  diagram,  or  Molier's  diagram  shows  that  be- 
tween two  given  temperatures  the  difference  of  heat  content  is  greater  at 
.a  greater  entropy.  An  example  is  given  in  Table  39,  the  values  being 
taken  from  Peabody's  entropy  table.  An  inspection  of  the  entropy  table 
also  shows  that  at  a  certain  entropy  the  heat  content  increases  at  a  nearly 
uniform  rate  for  considerable  intervals — of  20  or  even  40  degrees;  or 

TABLE  39 


Entropy  1.52 

Entropy  1.59 

Temperature 

Heat  content 

Temperature 

Heat  content 

228 
181 

1010.2 
953.0 

228 
181 

1037.7 

978.6 

57.2 

59.1 

AC/A£  is  practically  constant  for  this  interval  and  is  more  accurate  for 
the  wider  range. 

We  may  then  find  the  rate  at  which  the  available  heat  increases  with 
the  entropy  as  steam  expands  between  two  pressures  in  a  turbine  with  a 
certain  heat  factor  F.  The  calculation  may  be  made  at  some  intermediate 
point  between  Cj  and  C2,  where  the  heat  content  is  CA,  all  being  taken  at 


THE  STEAM  TURBINE  239 

the  same  entropy  cj>A.     It  is  preferable  for  the  sake  of  accuracy  to  have: 

CA  =  ^±^ 

as  nearly  as  possible  without  interpolation,  but  its  exact  value  must  be 
used  in  the  following  formulas. 

The  increase  of  entropy  in  expanding  from  PI  to  PA  will  be  due  to  the 
addition  of  the  heat  quantity : 

a         En    tn          r<  \ 
—  r  )    (l^i  —    L>A) 

The  heat  content  C*  at  entropy  0*  (at  constant  pressure  P,)  will  be: 

C*  =  CA  +  (1  -  F)  (C,  -  C,)  V33) 

from  which  <£*  may  be  found. 

Let  AC,  be  the  change  of  heat  content  for  the  temperature  interval 
AT7  at  constant  entropy  <f>A,  and  AC*  the  change  for  the  same  interval  at 
entropy  0*;  then  the  ratio  of  heat  change  is: 

AC* 

AT  =  AC* 
AC,  ~  AC, 

AT 

The  temperature  interval  should  be  equally  divided  on  either  side  of  TA. 
The  rate  of  increase  of  available  heat  between  Ci  and  C2  over  that  due 
to  adiabatic  expansion,  computed  for  heat  content  CA  is: 

AC*  _ 

AC,  "        _  Ci-  C2   /AC*         \  ,_, 

n r<~  ~  r       7*~  \A?r    "    /  ™' 

1^1  ^A  ^l  ^A    ^^^A  I 

Cl    —    C2 

If  this  gives  the  same  value  when  CA  is  taken  at  any  temperature 
between  Ci  and  C2,  the  variation  is  in  a  straight  line  for  equal  increments 
of  C;  as  this  is  very  nearly  true  for  saturated  steam,  the  fraction  of  increase 
of  available  heat  between  d  and  C2  will  be  one-half  the  value  given  by 
(34) .  Then  the  ratio  of  the  available  heat  to  that  available  due  to_a  single 
adiabatic  expansion  from  PI  to  P2  is: 


This  is  only  true  for  an  infinite  number  of  pressure  stages,  for  which  R 
is  equal  to  the  reheat  factor  FR;  but  it  may  be  used  for  constructing 
gram  which  is  generally  applicable. 


240 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


To  make  allowance  for  this  increase  of  available  heat  during  expansion, 
the  adiabatic  heat  assignments,  whether  equal  or  in  any  desired  propor- 
tion, must  decrease  from  C\  to  C2  at  the  same  rate  that  the  available 
heat  increases,  but  their  sum  must  not  change.  This  is  best  illustrated 
by  the  diagram  of  Fig.  144.  The  vertical  height  represents  Ci  —  C2, 
and  AC  the  desired  heat  assignment  per  stage,  being  a  fraction  of  Ci  —  C2 
which  is  the  adiabatically  available  heat  from  pressure  PI  to  P2. 

It  is  obvious  from  (34)  and  (35)  that  the  difference  between  the  hori- 
zontal dimensions  R  and  2  —  R  is  equal  to  the  rate  of  change  of  available 
heat,  and  that  their  average  is  unity.  The  dimensions  FD,  to  the  same 


*0 

•? 

f- 


f- 


f- 

vL 

\t 


--------  ,2-R 


H 


FIG.  144. 


scale  as  R,  are  the  distribution  factors,  each  one  bisecting  the  distance 
AC;  multiplying  AC  by  corresponding  values  of  FD  gives  the  heat  quan- 
tities which,  beginning  with  the  first  stage,  should  be  successively  sub- 
tracted at  constant  entropy  (corresponding  to  Ci),  in  order  to  determine 
the  heat  content  at  each  stage,  in  the  same  manner  that  equal  heat 
quantities  (Ci  —  Cz)/nP  were  subtracted  to  determine  the  pressures  for 
Fig.  143. 

An  inspection  of  Fig.  144  will  show  that  no  matter  how  the  heat  is 
distributed,  the  sum  of  the  -products  of  FD  and  AC  is  equal  to  the  sum  of 
AC.  For  equal  divisions,  Fm,  being  the  ratio  of  the  available  heat  to 
that  which  would  be  available  without  heat  degradation,  is  also  equal  to 
the  ratio  F/FS  which  is  the  reheat  factor  previously  mentioned.  It  is 
obvious  that  Fm  ( =  FR)  increases  with  the  number  of  stages,  its  limit 
being  the  distribution  ratio  R  for  an  infinite  number  of  stages,  and  this 
value,  found  by  another  method,  is  used  by  Martin  as  the  reheat  factor. 


THE  STEAM  TURBINE  241 

p 


(35a) 


s 


it  is  obvious  that  Fs  decreases  as  the  number  of  stages  increases,  and  this 
may  be  seen  in  Fig.  143.  It  then  follows  that  FR  increases  and  Fs  de- 
creases as  AC  becomes  smaller,  which  indicates  that  when  the  stages  are 
unequal,  FR  and  Fs  do  not  have  the  same  value  for  each  stage.  From 
Fig.  144,  as  FDi  =  FR,  it  may  be  seen  that  for  equal  stages  : 

Fa  =  1  +    l  -        ~~(R  -  1}  (36) 


It  may  then  be  assumed  that  for  any  work  division,  the  reheat  factor  for 
the  heat  drop  AC  is  given  by  (36). 

As  an  example  of  application,  the  pressures  of  Fig.  143  will  now  be 
determined  with  the  same  data  as  for  Fig.  141  and  a  heat  factor  of  0.6. 

Then  when  PI  =  164.8  and  0  =  1.56,  d  =  1193.3;  and  when  P2  = 
14.7  and  0  =  1.56,  C2  =  1018.0.  The  mean  heat  content  is: 


At  entropy  1.56  the  nearest  value  in  Peabody's  entropy  table  is  at  53.6  Ib. 
and  285  degrees,  for  which  C  =  1105.2  and  this  will  be  taken  as  CA,  at 
entropy  <f>A  (=  1.56).  From  (33): 

CB  =  1105.2  +  [0.4  X  (1193.3  -  1105.2)]  =  1140.44  B.t.u. 

At  temperature  285  degrees,  the  entropy  <f>B  at  which  this  value  is 
found  is  between  1.60  and  1.91;  interpolation  gives  <J>B  =  1.607. 

At  <£A  =  1.56,  taking  AT  at  40  degrees,  20  degrees  on  either  side  of 
285  degrees: 

AC  A  =  1127.7  -  1082.1  =  45.6 

At  <f>B  —  1.61,  for  20  degrees  either  side  of  285  degrees. 

AC*  =  1163.88  -  1116.36  =  47.52 
Then: 

AC.  _  47.52  _ 
ACA      45.60  " 

The  ratio  R  is  given  by  (35),  and  is,  when  CA  is  the  mean  between  Ci 
and  C2  : 

R  =  4£»  =  1.041  or  1.04  nearly. 
AC  A 

For  practical  application  a  portion  of  Fig.  144  may  be  omitted;  then 

16 


242 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Fig.  145  is  the  diagram  for  four  equal  stages  with  the  data  given  and 
determined. 

The  quantity  AC  is  the  same  for  each  stage  ;  or  : 

C,      1193-3       1018 


AC  = 


4 

TABLE  40 


=  43.825 


1 

Stage 

2 
Fz>-AC 

r3 

C'i-S(FD-AC') 
at  <f>  =  1.56 

4 
P 
(abs.) 

5 

Ci-SCF-AC) 

6 

<*> 

7 
Actual 
heat  drop 

8 

V 

Steam 

1993.30 

'    1193.30 

1193.30 

chest 

45.15 

164.80 

26.30 

1.560 

1148.15 

45.15 

1148.15 

1167.00 

1167.00 

1 

45.15 

44.25 

94.15 

26.30 

1.584 

1121.85 

4.590 

45.15 

1103.90 

1140.70 

1140.70 

2 

44.25 

43.40 

52.33 

26.30 

1.609 

1095.26 

7.852 

45.44 

1060.50 

1114.40 

1114.40 

3 

43.40 

42.50 

28.21 

26.30 

1.636 

1069.17 

14.306 

45.23 

4 

42.50 

1018.00 

14.70 

1088.10 

1.665 



25.070 

1.04- 


FOI-I.03 


Table  40  gives  the  steps  in  determining  intermediate  pressures,  and  a 
check  calculation  showing  actual  heat  drops  per  stage.  In  column  5, 
F-AC  subtracted  from  the  heat  content  of  each 
stage  gives  the  amount  of  heat  available  for 
adiabatic  expansion  in  the  following  stage.  The 
entropy,  column  6,  is  that  corresponding  to  the 
pressure  (column  4)  of  the  same  line. 

The  subtrahend  in  column  7  is  found  by 
following  down  the  entropy  given  in  the  same 
line  of  the  table,  to  the  pressure  given  in  the  line 
below. 

The  quantities  in  the  table  were  taken  from 
Peabody's  entropy  table  by  interpolation  and 
the  work  done  on  a  slide  rule.  The  heat  drop 
occurs  in  the  nozzles  connecting  the  compart- 
ments, and  which  really  belong  to  the  stage  related  to  the  compartment 
into  which  they  discharge, 


FIG.  145. 


THE  STEAM  TURBINE 


243 


An  alternative  method  which  possesses  some  advantages,  makes  use 
of  the  reheat  factor  instead  of  the  distribution  factor.  This  is  given  in 
Table  41.  In  column  2,  FR-&C,  for  stage  n  +  1,  is  subtracted  from  the 
heat  content;  the  resulting  heat  content  is  placed  hi  the  line  below,  and 
at  the  original  entropy  the  corresponding  pressure  is  found — sometimes 
by  interpolation.  The  unused  heat  of  this  stage  (1  —  FS)FR-AC  (=  FR- 
AC  —  F-AC)  is  now  added,  and  at  the  pressure  just  found  the  new  entropy 
is  taken.  FR-AC  of  the  following  stage  is  now  subtracted  and  the  re- 
sulting heat  content  down  this  new  entropy  line  locates  the  next  pressure, 
and  so  on. 

TABLE  41 


Steam 
chest 


1193.30 
45.15 

1148.15 
18.06 

1166.21 
45.15 

1121.06 
18.06 

1139.12 
45.15 

1093.97 
18.06 

1112.03 
45.15 

1066.88 


164.80 


94.15 


1.5600 


1.5840 


52.24 


27.83 


14.478 


1.6087 


1.6343 


The  specific  volume  for  any  pressure  used  for  determining  nozzle 
areas,  must  be  found  at  the  entropy  of  the  stage  above,  as  the  entropy 
of  each  stage  shows  fche  increase  due  to  heat  degradation  in  that  stage, 
and  this  occurs  after  the  steam  leaves  the  nozzles,  with  the  exception  of  a 
slight  amount  due  to  nozzle  friction.  This  applies  to  both  methods. 

The  check  of  this  latter  method  is  the  lowest  pressure,  and  had  FB- AC, 


244 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


been  more  accurately  taken  as  45.14,  the  resulting  pressure  in  Table  41 
would  have  been  more  nearly  14.7. 

The  value  of  R  is  found  as  before  from  (35),  and  FR  easily  calculated 
from  (36).  The  necessity  of  determining  many  values  of  F"D  for  multi- 
stage turbines  is  obviated  and  much  labor  saved. 

Approximate  values  of  R  are  given  in  Table  42  for  various  pressure 
ranges  and  heat  factors,  for  saturated  steam,  taken  from  Peabody. 

TABLE  42 


Heat  factor 

150  Ib.  gage  to  28  in. 
vacuum 

150  Ib.  gage  to  atmos- 
phere 

Atmosphere  to  28  in. 
vacuum 

0.55 

.09 

1.050 

1.040 

0.60 

.08       . 

1.045 

1.035 

0.65 

.07 

1.035 

1.030 

0.70 

.06 

1.030 

1.025 

0.75 

.05 

1.025 

1.020 

Superheated  Steam. — The  direct  method  of  pressure  distribution  just 
given  will  not  give  as  accurate  results  when  applied  to  superheated  steam; 
an  inspection  of  a  T$  Diagram  shows  this.  Assuming  the  nozzles  to  be 
designed  for  saturated  steam,  and  the  same  heat  factor  in  each  case,  the 
nozzles,  from  (11),  will  accommodate  less  superheated  steam,  but  from 
(3),  less  superheated  steam  is  required  for  the  same  power. 

The  entropy  diagram  for  superheated  steam  is  given  in  Fig.  146.  The 
dotted  lines,  as  before,  show  assumed  entropy  changes  due  to  nozzle 
friction,  while  the  shaded  areas  are  available  adiabatic  heat  drops  per 
stage.  With  less  initial  superheat,  the  lower  stages  would  be  like  those 
of  Fig.  143. 

For  any  pressures,  heat  drops  may  be  determined  as  by  the  method 
given  for  columns  5  to  7  of  Table  40.  This  assumes  that  the  nozzle  areas 
are  determined  for  these  pressures,  heat  drops  and  specific  volumes,  and 
that  the  actual  heat  factor  is  the  value  used  in  the  calculations. 

Velocity  Due  to  Reheating. — Early  in  the  present  paragraph  it  was 
assumed  in  Formula  (31)  that  the  heat  available  for  producing  velocity  is 
(Ci  —  C2)/nP.  Later  it  was  shown  that  heat  drop  along  the  new  entropy 
due  to  reheating  is  the  available  heat;  and  while  a  smaller  percentage  of 
this  is  converted  into  work  (as  discussed  under  Effect  of  Heat  Factor), 
there  seems  to  be  no  reason  why  it  should  not  be  available  for  producing 
velocity  if  we  assume  that  all  of  the  energy  of  residual  velocity  is  used  for 
reheating.  With  this  assumption  (Ci  —  C2)/nP  should  be  multiplied 


THE  STEAM  TURBINE 


245 


by  FD,  or,  more  generally,  each  part  of  Ci  —  C2,  or  AC,  for  any  distribu- 
tion, should  be  multiplied  by  the  reheat  factor  FB  for  that  stage,  the  quan- 
tity FR-&C  being  the  heat  drop  along  the  entropy  found  for  a  given 
stage.  The  quantity  AC  is  the 
desired  fraction  of  the  total  heat 
drop  before  being  multiplied  by 
the  distribution  factor  FD. 

For  saturated  steam  FR  may 
be  found  from  (36)  for  any  work 
division.  The  formula  for  velocity 
then  becomes: 


V  =  223.7V '(1  ~  yW*'&C  (37) 
The  sum  of  the  products  FR-AC 
is  greater  than  Ci  —  €2,  but  the 
unused  heat  in  the  stages  of  a 
multi-stage  turbine  is  available  to 
cause  flow  intothe  following  stages, 
while  in  the  single-stage  turbine 
the  surplus  energy  due  to  residual 
velocity  is  discharged  at  the  lo  vver 
pressure  limit  and  is  no  longer 
available.  The  actual  work  done, 
however,  is  the  product  of  Ci  —  C2 
and  F,  so  that  apparently  the 
velocity  increase  is  not  available 
for  increased  work;  in  view  of  this 
it  may  perhaps  be  permissible  to 
ignore  FR  in  drawing  velocity  dia- 
grams and  in  area  calculations, 
or  to  make  some  allowance  for  it 
in  choosing  the  friction  factor  y. 

General   Dimensions.  —  Equat- 
ing (11)  and  (13)  gives: 


kmD2  sin  a  =  144w 
from  which: 
D  =  6.77 


wv 


sn  a 


(38) 


(39) 


FIG.   146. 


where  D  is  the  diameter  in  inches  at  pitch  circle  of  blades,  guides  or  noz- 
zles. In  determining  conditions,  or  analyzing,  it  may  also  be  desirable  to 
solve  for  other  quantities  in  (38). 


246  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

With  impulse  turbines  the  value  of  m  for  maximum  ratio  of  blade 
length  to  diameter  of  pitch  circle  will  give  the  minimum  wheel  diameter 
for  the  last  stage;  this  is  found  by  multiplying  this  ratio  by  the  ratio 
di/dz  found  from  (19).  Then  V  and  v  are  for  the  discharge  from  the 
last  nozzles.  F  may  depend  upon  S,  which  has  been  previously  fixed, 
or,  vice  versa,  their  relation  being  given  by  the  velocity  diagram.  Then 
N,  the  r.p.m.,  is  given  by: 

irDN  S 

12  X  60  =  9  D 

Some  limiting  values  of  m  were  given  in  Par.  86. 

According  to  Martin,  in  a  pressure-stage  turbine  with  single  velocity 
stages,  with  a  28-in.  vacuum,  D2  should  not  be  less  that  0.57  X  output  in 
kw. 

The  relation  between  diameter,  speed,  number  of  stages  and  efficiency 
is  mentioned  in  Par.  95. 

Multiplying  (38)  by  S  and  substituting  the  value  of  w  in  terms  of  (3) 
gives: 


3 

D  =  19.5  3 


kmNF(Ci  -  C8)  sin  a 


kwVV 
kmeMNF(Cl-C2)sma 

The  value  of  v  (or  x,  from  which  v  may  be  computed)  may  be  found 
at  the  pressure  within  the  stage  considered,  and  at  the  value  of  heat 
content  equal  to  : 

Cx  =  Cn  +  (1  -  F)  (Ci  -  Cn)  -  (1  -  Fs)FR'AnC          (40a) 

The  subscript  n  refers  to  any  stage;  Cn  is  the  heat  content'at  the  pressure 
considered,  at  the  same  entropy  as  Ci,  and  AnC  is  the  heat  drop  for  the 
stage,  along  the  same  entropy.  It  is  accurate  enough  for  most  purposes, 
especially  with  reaction  turbines,  to  take  F  instead  of  Fs  and  omit  FB; 
then 

Cx  =  Cn  +  (1  -  F)  (Ci  -  Cn  -  AnC)  (406) 

Formula  (40)  is  convenient,  as  the  effect  of  the  various  factors  may  be 
readily  seen. 

Number  of  Equal  Stages.  —  As  calculations  to  determine  the  number  of 
stages  seldom  results  in  a  whole  number,  the  influence  of  reheating  will 
be  neglected  and  the  simpler  formulas  used  Two  factors  influence  the 
number  of  pressure  stages:  first,  the  form  of  nozzles,  and  second,  the 


THE  STEAM  TURBINE  247 

peripheral  speed.  If  convergent  nozzles  are  to  be  used,  the  heat  drop 
AC  in  any  stage  must  be  no  greater  than  that  due  to  a  pressure  drop  to 
0.58  of  the  pressure  in  the  preceding  stage;  then: 

—   Gl    —    G  2  f  n\ 

nr  ?  —^-  (41) 

With  less  pressure  stages  and  a  greater  heat  drop,  divergent  (con- 
vergent-divergent) nozzles  must  be  used. 

If  S  is  the  limiting  factor,  V  may  be  found  after  the  velocity  diagram 
is  designed;  then  solving  for  nP  in  (31)  gives: 

-         OU,UUU        /., 


The  actual  value  of  AC,  or  of  V  and  S  may  then  be  determined  if  nP  is 
taken  at  the  nearest  whole  number  which  will  not  exceed  the  limits 
imposed.  . 

Unequal  wheel  diameters  are  sometimes  employed  to  avoid  extreme 
blade  lengths.  This  leads  to  unequal  stages  and  sometimes  to  the  reduc- 
tion of  the  number  of  stages.  If  the  same  form  of  velocity  diagram  is 
used  in  each  stage,  V  is  proportional  to  S,  which  is  proportional  in  turn  to 
D;  then  if  nozzle  friction  is  assumed  the  same,  the  heat  drop  is  propor- 
tional to  D,  and  for  any  stage,  with  these  assumptions  : 


d-C2 

D  may  be  determined  by  (39)  for  the  last  stage  by  assuming  V,  and 
the  design  worked  along  stage  by  stage  toward  high-pressure  end,  re- 
ducing wheel  diameters  at  discretion  and  checking  by  (38)  for  blade 
lengths.  Assuming  D  and  S,  V  may  be  calculated  tentatively,  and  heat 
drop  AC  may  be  found  from  (37),  neglecting  FR;  then  from  (36),  FR 
may  be  found,  from  which  V,  S  and  D  may  be  corrected,  using  (37)  for 
calculating  V.  The  corrected  relation  bewteen  heat  drop  and  pitch 
diameter  would  then  be,  for  similar  velocity  diagrams: 

FR(l  -  ?/)AC  D*_ 

2[FR(l  ~  2/)AC]       SZ)2 
but  the  error  is  slight. 

If  convergent  nozzles  are  to  be  used,  divide  the  absolute  condenser 
pressure  by  0.58  to  obtain  the  absolute  pressure  in  the  preceding  stage; 
the  heat  drop  AC  between  these  pressures  will  be  that  for  the  last  stage, 
and  V  may  be  found  from  (37)  instead  of  assumed.  This  will  determine 
S,  which  may  be  greater  or  less  than  desired  and  a  compromise  must  be 
made. 


248  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Another  method  is  to  start  from  the  high-pressure  end  with  the 
assumption  of  equal  stages  and  equal  diameters.  If  toward  the  low-pres- 
sure end  the  blades  are  too  long,  the  wheel  diameter  may  be  increased,  V 
increasing  in  proportion,  until  m  in  (38)  is  not  too  large;  if  the  form  of 
velocity  diagram  is  not  changed,  D/V  is  constant  and  m  varies  inversely 
as  D3.  A  greater  heat  drop  will  accompany  an  increase  in  diameter  and 
the  remaining  stages  must  reduce  the  heat  content  to  exactly  Cz.  The 
values  of  AC  should  check  with  (43)  for  similar  velocity  diagrams;  they 
may  then  be  laid  off  on  Fig.  144  and  the  design  completed. 

In  some  of  the  modern  large  multi-stage  turbines  the  wheels  increase 
in  diameter  from  the  first  stage  to  the  last.  There  is  a  single  velocity 
stage  in  each  pressure  stage,  with  full  peripheral  admission.  The  work 
per  stage  necessarily  increases  toward  the  low-pressure  end  of  the  turbine. 

Conservation  of  Residual  Velocity. — The  higher  stages  of  most  pressure- 
stage  turbines  have  partial  admission;  that  is,  steam  is  admitted  through 
but  a  portion  of  the  periphery.  The  reason  for  this  is,  that  otherwise 
extremely  short  blades  are  required  to  keep  the  nozzle  area  to  the  required 
amount.  The  blades  are  then  kept  an  appreciable  distance  from  the 
next  set  of  nozzles  in  order  to  allow  the  steam  to  spread  out  and  fill  the 
nozzles.  With  admission  around  the  entire  periphery,  equal  wheel 
diameters,  and  clearance  on  the  discharge  side  reduced  to  about  one-half 
the  blade  width,  a  good  percentage  of  the  residual  velocity  is  available 
as  initial  velocity  of  entrance  to  the  nozzles.  For  any  stage  thus  situated 
the  available  energy  is : 


^C  (44) 

where  z  is  the  fraction  of  residual  energy  available  as  kinetic  energy,  and 
VR  is  the  residual  velocity  of  the  preceding  stage. 

The  value  of  F  should  be  nearly  1  +  /*  times  its  value  without  con- 
servation of  residual  energy  and  this  slightly  reduces  FR,  making  it 
difficult  to  arrive  at  exact  values.  The  fraction  of  increase: 

.C  (45) 


^223.7y 
may  be  determined  approximately  by  taking: 

AC  =  £LZ_£» 


THE  STEAM  TURBINE  249 

or  dividing  unequally  along  the  adiabatic  as  previously  described,  as 
VR2  will  have  the  same  ratio  to  FR-AC  as  when  the  correction  is  made. 

In  order  to  have  V  the  same  in  each  stage  the  available  energy  must 
be  the  same.  For  the  stages  in  which  there  is  no  conservation  —  the 
stages  with  partial  admission  —  let: 


This  must  equal  the  value  given  by  (44)  ;  as  the  stages  are  so  nearly  equal, 
F'R  may  be  taken  as  equal  to  FR,  and  y  may  be  assumed  the  same  for 
each  stage.  Then  : 

A'C  =  (1  +  /i)  AC  (46) 

If  UP  and  n'p  denote  the  number  of  stages  with  and  without  conserva- 
tion respectively,  it  is  obvious  that: 


Then: 


\n  i  —      z  ,An^ 

AC  =  —TT\  —  i  —  \~~i  —  0*7) 

riP(l  +  fj.)  +  nP 


Taking  A'C  from  (46),  a  diagram  similar  to  Fig.  144  may  be  constructed, 
and  the  proper  assignments  per  stage  made. 

For  the  value  of  z  in  (44),  Peabody  gives  0.9;  Martin  gives  0.4  to  0.5 
when  the  clearance  between  the  wheels  and  diaphragms  is  reduced  to  a 
minimum,  and  the  entrances  to  the  guide  blades  are  properly  formed  to 
receive  the  discharge  from  the  preceding  wheel.  When  the  stages  are 
not  nearly  equal,  or  part  of  them  contain  velocity  stages,  the  calculation 
becomes  involved;  trial  and  error  must  be  resorted  to  and  check  calcula- 
tions made. 

The  foregoing  discussion  will  give  some  idea  of  the  conservation  of  the 
energy  of  discharge;  aside  from  increasing  the  heat  factor  between  5  and 
10  per  cent.,  it  is  probably  permissible  to  arrange  blading  to  take  advan- 
tage of  conservation  but  neglect  it  in  calculation. 

91.  Pressure-velocity-stage  impulse  turbines  have  two  or  more 
velocity  stages  in  one  or  more  of  the  pressure  stages.  The  simplest  case 
is  the  stationary  Curtis  turbine  built  for  small  powers  by  the  General 
Electric  Co.  There  are  usually  two  pressure  stages,  with  two,  and  some- 
times three  velocity  stages  in  each.  In  this  turbine  the  rim  velocity 
varies  inversely  as  the  number  of  velocity  stages  and  as  the  square  root 
of  the  number  of  pressure  stages.  Due  to  fewer  pressure  stages  the  heat 


250  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

drop  per  stage  is  so  large  that  diverging  nozzles  are  required.  The 
velocity  diagram  for  each  stage  is  similar  to  Fig.  140  for  two  velocity  stages, 
from  which  V  may  be  measured  or  calculated  for  a  desired  value  of  S. 
Then  AC  for  each  stage  may  be  calculated  from  (37),  and  the  required 
number  of  pressure  stages  from  (41)  ;  the  design  may  then  proceed  as  for 
any  pressure-stage  turbine  with  equal  pressure  stages. 

A  not  uncommon  arrangement  is  to  place  two  velocity  stages  in  the 
first  pressure  stage  only,  reducing  the  pressure  and  the  number  of  stages. 
The  velocity  diagrams  may  be  designed  and  the  values  of  V  obtained. 
Sometimes  S  may  be  the  same  for  the  one-stage  and  two-stage  diagrams, 
but  often  the  two-stage  wheels  are  made  smaller  —  though  sometimes 
larger  —  in  diameter;  in  any  case  V  is  greater  for  the  first  stage  and  the 
heat  drop  AC  will  be  greater.  In  general  : 

'  (48) 


Cl  -  C2 

the  subscript  n  denoting  any  stage.  272  may  be  solved  for,  and  sub- 
tracting successively  V\  V2Z)  etc.,  the  number  of  stages  may  be  deter- 
mined. The  distribution  factors  may  be  neglected  for  the  present,  the 
pressures,  specific  volumes  and  areas  being  approximate;  then  the  blade 
lengths  may  be  checked  by  (38)  as  .the  calculating  proceeds  toward  the 
low-pressure  end,  and  an  increase  in  diameter  may  be  desirable.  It  may 
be  necessary  to  increase  the  diameter  of  the  last  one  or  two  wheels  to  have 
SF2  entirely  accounted  for.  This  is  equivalent  to  the  second  method 
under  unequal  wheel  diameters  in  the  preceding  paragraph,  and  is  more 
general.  As  previously  stated,  in  finding  AC  from  (37),  FR  may  first  be 
neglected,  but  from  the  value  so  found,  FR  may  be  obtained  from  (36) 
and  AC  recalculated  by  (37)  with  sufficient  accuracy. 
As  with  Formula  (43),  (48)  is  more  correctly  stated: 
FB(1  -  y)knC]  Vn2 


The  method  just  outlined  is  applicable  to  any  arrangement,  such  for 
instance,  as  the  Curtis  marine  turbine  referred  to  in  Par.  12,  Chap.  IV. 
When  the  approximate  calculation  is  complete  and  the  number  and  pitch 
diameter  of  the  wheels  determined,  the  values  of  AC  may  be  laid  off  on  a 
diagram  such  as  Fig.  144  and  the  nozzle  and  blade  dimensions  checked. 
In  the  preliminary  calculations,  nozzle  exits  only  need  be  determined. 

92.  Trajectory  and  Lead.  —  The  actual  path  of  the  steam  in  its  pas- 
sage through  the  moving  vanes  of  a  turbine  is  known  as  the  trajectory. 
The*  relative  velocity  of  steam  VN  in  impulse  blading  is  constant  if  we 
neglect  the  effect  of  friction,  which  for  simplicity  will  be  done  in  the 
present  discussion. 


THE  STEAM  TURBINE 


251 


Let  the  concave  surface  of  a  blade  in  Fig.  147  be  divided  into  a  number 
of  parts  and  lettered  a,  b,  c,  etc.  When  the  steam  of  velocity  VN  has 
passed  over  the  distance  ab,  the  blade  has  moved  the  distance  W  due 
to  its  velocity  £;  then: 

66'  =  ab  •- 


cc' 


=  (ab  +  be)  ~ 

V  N 


FIG.  147. 


FIG.  148. 


and  so  on  until: 


eer 


ae 


_ 

VN 


(49) 


is  the  distance  the  blade  travels  in  the  time  it  takes  a  particle  of  steam 
to  pass  over  its  surface.  The  distances  ab,  ae,  etc.,  must  be  measured 
along  the  curve;  a  flexible  scale  may  be  used  for  this,  or  a  close  approxima- 
tion may  be  made  by  small  divisions  on  a  large  scale  drawing. 

Trajectories  are  shown  in  Fig. 
148  for   a  number  of  ratios  of  S        — 
to  V. 

The  convex  side  of  the  blade  is 
different  in  form  and  gives  another 
path;  by  tracing  the  trajectories  of 
both  sides  as  in  Fig.  149,  some  idea 
may  be  obtained  of  the  path  of  the 
stream  as  it  passes  between  the  FIG  14Q 

blades.     An   example   of   a   simple 

wheel  in  which  a  =  20  and  S  =  N  cos  a/2,  is  given  in  Fig.  149,  and 
for  a  2-velocity-stage  wheel  in  which  S  =  V  cos  a/5  in  Fig.  150. 

Lead. — In  pressure  stages  with  partial  admission  in  which  there  is 
considerable  space  between  the  wheel  and  the  next  set  of  nozzles,  the 
residual  energy  of  the  jet  is  dissipated  and  there  is  little  or  no  initial  velo- 
city of  entrance,  so  it  makes  little  difference  where  the  succeeding  noz- 


252  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

zles  are  located;  but  if  the  clearance  is  small,  so  that  part  of  this  velocity 
might  be  utilized,  the  nozzles  should  be  placed  so  as  to  receive  this  re- 
sidual steam  with  as  little  loss  as  possible.  This  may  be  done  by  finding 
the  distance  passed  over  as  the  steam  passes  through  the  blades  as  just 
described,  and  placing  the  next  set  of  nozzles  so  as  to  receive  this  dis- 
charge; as  this  set  usually  subtends  a  larger  arc  than  those  preceding,  but 
little  difficulty  is  experienced.  Where  there  are  two  or  more  velocity 
stages  with  partial  admission,  it  is  absolutely  necessary  for  the  guides  to 


FIG.  150. 

be  so  placed  that  all  of  the  steam  discharged  from  one  wheel  will  be 
directed  to  the  next;  it  is  well  in  this  case  to  provide  one  or  two  extra 
guides  at  each  end  of  the  row.  For  marine  turbines  run  at  variable  speeds, 
enough  additional  guides  must  be  provided  for  the  extremes  of  speed. 

In  using  formulas  for  finding  the  trajectory,  the  average  velocity 
would  probably  be  more  correct.  This  may  be  found  by  assuming  the 
friction  loss  uniform  throughout  the  passage.  Between  entrance  and 
exit  the  fraction  of  loss  is  1  —  #;  at  the  point  b  it  is: 

ab  f  , 

-  (1  —  0) 

ae 


THE  STEAM  TURBINE  253 

in  which  ae  is  the  total  length  of  the  passage  measured  along  the  surface 
of  the  blade.     The  velocity  coefficient  between  a  and  b  is  then: 


and  the  velocity  at  b  is: 

-  -'•'•  '•* 

The  average  velocity  between  a  and  b  is 


FIG.  151. 

Then: 


(50) 


The  distance  cc  may  be  found  by  using  ac  instead  of  ab,  and  in  like 
manner  other  points  on  the  curve  may  be  found.  At  exit,  where  ab  is 
changed  to  ae: 


Formula  (50a)  may  be  used  for  finding  the  lead.  Fig.  151  shows  the 
arrangement  of  nozzles  and  guide  blades  for  two  velocity  stages,  the 
dotted  lines  showing  the  theoretical  limits  of  the  steam  path.  The  lines 
from  nozzles  to  blades  follow  the  direction  of  the  stream  from  the  nozzles, 
while  from  moving  blades  to  guides  the  direction  is  tangent  to  the  trajec- 
tory curves  at  exit. 


254 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Due  to  variable  velocity  of  steam  in  reaction  blading,  it  is  not  practi- 
cable to  find  the  trajectory;  and  as  the  passages  form  convergent  nozzles 
in  which  expansion  occurs,  and  there  is  full  admission,  it  is  not  important. 

It  is  desirable  to  draw  trajectory  curves  for  both  front  and  back  sur- 
faces of  the  passage  for  all  moving  blades.  If  the  curves  are  not  smooth, 
and  free  from  too  sudden  changes,  the  blade  form  should  be  modified  if  the 
best  efficiency  is  desired. 

If  friction  is  neglected,  (50)  may  be  written  : 


__ 
ab        V* 

Then  it  is  obvious  from  Fig.  133  that  for  the  convex  side  of  the  blade  the 
trajectory  along  the  straight  portion  at  entrance  will  be  parallel  to  the 
jet  from  the  nozzle.  If  friction  is  considered,  (50)  may  be  written: 

bV_  =  _  S 

ab  /.       ab    1  — 


Then  in  order  that  the  trajectory  at  entrance  may  be  parallel  to  the  jet  so 
as  not  to  have  a  retarding  influence  on  the  back  of  the  blade,  W  must  be 
greater,  making  the  entrance  angle  greater  than  6  as  shown  in  Fig.  135. 

93.  The  Reaction  Turbine. 
The  essential  feature  of  the 
reaction  turbine  is  the  ex- 
pansion of  steam  in  the  mov- 
ing blades.  The  guides  cor- 
respond to  the  nozzles 
between  stages  in  a  pres- 
sure-stage impulse  turbine 
and  present  no  new  feature. 
Although  the  principle  is 
simple,  the  large  number  of 
stages  makes  a  detailed 
analysis  difficult  due  to  the 
resulting  accumulation  of  error  if  correct  coefficients  are  not  used.  It 
has  been  stated  that  the  design  of  reaction  turbines  is  based  upon  empirical 
rules  fixed  by  experience,  to  a  larger  extent  than  for  other  turbines.  Be 
that  as  it  may,  considerable  deviation  from  the  conditions  indicated  by 
theory  is  practised  in  the  interest  of  simplified  construction,  with  appar- 
ently no  deleterious  effect  upon  efficiency.  The  same  general  method 
will  be  used  as  in  the  study  of  the  impulse  turbine,  with  certain  modi- 
fications found  desirable  for  this  type. 


FIG.  152. 


THE  STEAM  TURBINE  255 

Reaction  Bidding. — The  velocity  diagram  is  similar  to  that  of  the 
impulse  turbine  as  shown  in  Fig.  152.  Vx  is  always  greater  than  VN  due 
to  expansion  between  the  blades  caused  by  heat  drop  AC.  The  effect 
of  friction  may  not  well  be  shown  on  the  diagram. 

If  Fig.  152  is  for  the  first  stage,  V  is  due  to  the  heat  drop  only,  but  if 
for  any  following  stage  it  is  due  to  the  heat  drop  and  also  the  residual  ve- 
locity VR  from  the  preceding  row  of  blades.  Likewise  the  exit  velocity 
Vx  from  the  moving  blades  is  due  to  the  relative  velocity  VN  at  entrance, 
and  the  expansion  due  to  the  heat  drop  AC. 

As  already  stated  in  Par.  84,  the  angle  of  discharge  of  both  blades  and 
guides  depends  upon  the  pitch  and  setting  of  the  blades  and  is  not  obvious 
from  the  drawing,  but  that  has  no  special  bearing  upon  our  calculations, 
as  the  blades  are  assumed  to  be  set  to  secure  the  angles  desired,  .and 
shown  upon  the  velocity  diagram. 

The  forces  acting  upon  the  blades  are  determined  as  in  Par.  86  for  the 
impulse  turbine,  and  are: 


and: 


an 

fw  =  -(V  cos  a  +  Vx  cos  0  -  S)  (51) 

t7 

fr  =  —(V  sin  a  —  Vx  sin  0)  (52) 

For  any  but  the  first  set  of  guides — which  are  really  the  first  set  of 
nozzles— let  A^C  be  the  heat  drop,  and  let  ABC  be  the  heat  drop  for  the 
following  row  of  blades  including  reheating;  then  equating  energy: 


and: 


where  i  and  j  are  coefficients  making  allowance  for  losses. 

It  is  convenient  to  consider  one  row  each  of  guides  and  blades  as  a 
stage;  then  for  one  stage,  the  available  heat  is: 


F  -AC=A  r  i  AC-  . 

*R  AC  h  AcC  50,000  i 

It  is  usual  to  assume  that  Vx  =  V,  and  VR  =  VN;  it  then  follows 
that  0  =  a,  and  <5  =  0.     Then  (53)  becomes  : 


which  is  the  heat  drop  per  stage  (for  one  row  each  of  guides  and  blades). 


256  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Also  (51)  and  (52)  become: 

fw=-  (2V  cos  *-S) 


(55) 
and: 

f,=  0 

From  Fig.  152: 

TV  =  V2  sin2  a  +  (V  cos  a  -  S)2 

(66a) 

Then  the  energy  supply  in  foot-pounds  per  second  received  by  each 
stage  is: 


Es=  778^«-AC  =  l-jl--2cosa-  (56) 

As  mentioned  in  connection  with  impulse  turbines,  FR  may  be  neglec- 
ted in  preliminary  calculations,  and  it  is  not  altogether  clear  whether 
values  of  i  and  j  given  by  Martin  were  based  upon  FR-AC  or  AC;  if  the 
latter,  FR  may  be  neglected. 

The  work  done  in  foot-pounds  per  second,  per  stage  is: 


Ew  =  fwS=  (2  cos  a  -     )  (57) 

The  diagram  efficiency  is  then: 


S 
Ew 


(58) 


More  general  values  of  fw  and  AC  given  by  (51)  and  (53)  may  be  used 
in  (57)  and  (56)  for  determining  eD,  but  (58)  gives  a  good  idea  of  the 
relative  effects  of  different  values  of  a  and  S/V. 

Martin  gives  i  =  0.9  and  j  =  0.52,  which  were  found  by  analyzing 
the  performance  of  a  large  high-pressure  marine  turbine,  but  he  adopts 
the  values: 

i  =  0.89  and  j  =  0.5. 

As  many  reaction  turbines  have  open-ended  blades  and  guides,  there 
is  tip  leakage,  and  only  a  portion  of  the  steam  does  useful  work.  Fig. 
153  is  a  diagram  showing  the  arrangement  of  blades  and  guides,  d  being 


THE  STEAM  TURBINE 


257 


the  blade  length  and  c  the  tip  clearance.  The  flow  through  passages  of 
this  kind  presents  a  very  complicated  problem  and  the  fraction  of  the 
steam  passed  through  which  may  be  assumed  to  be  effectively  applied  is 
not  obvious;  but  Martin  gives  it  as: 

d  -  c 


d+\c 


</////////////////////////.  fa.si. 


o 

A 

f 

1 

VJ 

:* 

i 

V) 

'//////////////////////// Drum 


where 


FIG.  153. 
1 


sin  a 


He  gives  for  normal  blades,  X  =  2.15  (a  =  18°  —  30');  for  semi- wing 
blades,  X  =  1.1  (a  =  28P  30');  and  for  wing  blades  X  =  0.6  (a  =  38°  30'). 
The  efficiency  of  the  blading  is  then : 

d-  c 


=  CD 


(59) 


Neglecting  leakage  from  shaft  glands  and 
other  packings,  the  heat  actually  converted 
into  work  is: 


If  eB  and  FR  are  assumed  constant,  it  is  ap- 
parent that: 


F  =  eBF 


(60) 


The  average  value  of  eBFR  may  be  taken, 
and  (60)  used  to  obtain  an  approximate  value 
of  the  total  heat  factor.  The  loss  by  leakage 
is  about  4  or  5  per  cent.,  and  this  may  be 
deducted  from  the  value  found  by  (60).  As 
FR  =  F/Fs,  (60)  reduces  to  Fs  =  eB,  which  is 

.  nearly  true  for  the  reaction  turbine,  as  eB,  the  blade  efficiency  accounts 
for  all  losses  except  gland  leakage. 

17 


V///77/ 


FIG.  154. — Shrouded  blades. 


258  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  blading  of  some  reaction  turbines  is  shrouded,  as  shown  in  Fig. 
154,  with  the  idea  of  preventing  tip  leakage.  Jude  says  that  if  shrouded 
blades  or  guides  have  their  roots  raised  above  the  level  of  the  drum  or 
casing,  the  remedy  is  worse  than  the  evil,  unless  such  construction  allows 
a  working  clearance  of  less  than  one-half  of  that  used  in  the  ordinary 
construction. 

Work  Division. — As  explained  in  Chap.  IV,  the  reaction  turbine  con- 
sists usually  of  several  cylinders,  and  upon  each  cylinder  are  several  groups, 
or  barrels,  each  of  which  are  made  up  of  several  rows  of  moving  blades  or 
stages.  The  blades  of  a  group  are  all  of  the  same  length,  although  theo- 
retically they  should  increase  gradually  to  allow  for  the  increased  volume. 
The  pressure  drop  in  each  row  is  so  small  that  the  ratio  of  final  to  initial 
pressure  is  always  greater  than  0.58;  therefore  the  passages  should  be 
converging,  as  explained  in  Par.  84,  and  even  though  the  theoretical  in- 
crease in  blade  length  were  effected  in  practice,  the  length  of  each  blade 
or  guide  would  be  constant. 

Fig.  28,  Chap.  IV,  shows  a  drum  divided  into  cylinders  and  barrels.  It 
is  obvious  that  at  entrance  to  the  first  stages  of  each  cylinder,  and  in  some 
cases  of  the  different  groups,  there  will  be  no  initial  velocity,  and  a  treat- 
ment similar  to  that  in  the  preceding  paragraph  for  conservation  of 
residual  velocity  may  be  applied,  allowing  a  greater  heat  assignment  to 
these  stages.  In  view  of  the  large  number  of  stages,  the  difference  is 
insignificant  and  will  be  neglected. 

The  blades  of  a  group  are  usually  identical  in  form  and  setting,  but 
sometimes  the  blades  are  " gaged"  so  as  to  keep  the  velocity  of  discharge 
constant  throughout  the  group.  The  gaging  consists  in  setting  the 
blades  for  a  larger  discharge  angle;  this  gives  an  increased  area  as  shown 
by  (12),  thus  fulfilling  the  theoretical  requirement  which  would  also  be 
obtained  by  increasing  blade  length  with  a  constant  discharge  angle. 
In  the  present  discussion  angles  will  be  assumed  constant,  or  an  average 
value  will  be  assumed  for  a,  and  calculations  for  height  of  blades  will  be 
at  the  center  of  the  group,  or  for  the  average  height  if  the  blades  were 
assumed  to  progress  in  height  as  called  for  theoretically. 

The  values  of  a.  and  S/V  are  selected  to  give  a  good  diagram 
efficiency,  and  the  values  of  S  for  the  largest  and  smallest  cylinders  are 
usually  determined.  The  heat  drop  per  stage  may  be  found  for  any 
group  from  (54),  neglecting  FR.  Let  subscripts  1,  2,  etc.,  denote  the 
different  groups,  beginning  with  the  high-pressure  end,  the  groups  being 
arranged  on  the  different  cylinders.  Then  if  n  is  the  number  of  stages 
in  a  group : 

wrAiC  +  rc2-A2C  +  na-A3C  ....  =  Ci  -  C2  (61) 


THE  STEAM  TURBINE 


259 


The  groups  may  be  arranged  tentatively  and  calculations  started  from 
either  end. 

If  there  are  three  cylinders,  which  is  common  in  practice,  the  work 
may  be  equally  divided  between  them,  or  approximately  so;  Martin 
says  that  the  high  and  intermediate  cylinders  each  do  J£  °f  the  work, 
and  the  low-pressure  cylinder  the  rest.  Any  division  of  work  may  be 
assumed  and  the  heat  quantity  Sn-AC  for  each  cylinder  may  be  laid  off 
on  a  diagram  such  as  Fig.  144.  Then  the  groups  may  be  arranged  for 
each  cylinder.  As  the  increase  of  volume  is  slower  in  the  upper  stages, 


FIG.  155. 

the  number  of  rows  per  group  would  naturally  be  greater  at  the  high- 
pressure  end,  decreasing  toward  the  low-pressure  end,  but  this  is  appar- 
ently not  always  so  in  practice. 

When  the  adiabatic  heat  drop  Ci  —  C2  has  been  arranged,  the  appor- 
tionment diagram  may  be  something  like  Fig.  155,  which  is  assumed  to 
give  equal  work  for  each  cylinder.  Two  barrels  per  cylinder  are  shown, 
but  there  are  usually  more. 

It  was  stated  that  blade  heights  would  be  determined  at  the  center  of 
each  group,  or,  at  half  the  heat  drop.  It  is  possible  that  some  other  heat 
drop  might  be  preferable,  but  when  this  is  determined,  it  becomes  neces- 
sary to  find  the  pressures  at  these  points  in  order  to  find  the  specific 


260  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

volumes  to  be  used  in  determining  blade  heights.  Assuming  the  center 
of  the  group — although  any  other  point  would  offer  no  difficulty — the 
first  heat  drop  is: 

A       ni-AiC 
"~2~ 

Then  C\  —  FDAA  gives  the  heat  content  at  the  center  of  the  first  group 
at  the  original  entropy,  and  the  pressure  PI  may  be  found.  FD  is  taken 
at  the  center  of  the  heat  drop  as  explained  in  Par.  90.  FDBB,  FDCC,  FDDD, 
etc.,  may  be  subtracted  in  succession  and  the  pressures  at  the  center 
of  each  group  determined.  Then  the  new  entropies  and  specific  volumes 
may  be  found  as  described  in  Par.  90,  and  the  necessary  area  found 
from  (11),  using  the  value  of  V  already  determined.  Blade  height  d 
may  be  found  from  (12),  taking  k  =  1,  or  m  may  be  found  from  (13), 
and  this  should  be  within  practical  limits,  which  will  be  considered 
presently.  . 

Disc-and-drum  Type. — Various  combinations  of  impulse  and  reaction 
turbines  are  built,  and  as  discs  are  generally  used  for  the  former  and 
drums  for  the  latter,  the  above  name  is  sometimes  used.  In  some  there 
is  an  impulse  pressure  stage  containing  two  velocity  stages,  followed  by  a 
large  number  of  reaction  stages,  sometimes  divided  into  two  parts,  form- 
ing a  double-flow  turbine.  The  Tosi  marine  turbine  described  by  Martin 
has  six  impulse  pressure  stages,  the  first  containing  four  velocity  stages, 
the  remainder  three;  this  is  followed  by  a  reaction  drum  containing  four- 
teen stages — or  rows  of  moving  blades. 

From  what  has  preceded,  no  detailed  method  of  design  need  be  given 
for  this  type;  the  procedure  may  be  similar  to  the  design  of  the  pressure- 
stage  turbine  with  velocity  stages  in  the  first  stage. 

94.  Practical  Notes  on  Reaction  Turbines. — Few  cases  arise  in  turbine 
design  in  which  conditions  may  be  fixed  by  the  application  of  theoretical 
formulas;  many  preliminary  calculations  and  alterations  must  be  made 
before  the  final  design  is  complete.  However,  empirical  rules  are  formu- 
lated during  the  progress  of  an  art  which  greatly  facilitate  design,  and  a 
few  suggestions,  derived  from  various  sources  will  be  given. 

From  (58)  it  may  be  found  that  the  highest  diagram  efficiency  is 
obtained  when  S/V  is  about  0.9,  when  the  values  of  i  and  j  given  by 
Martin  are  used.  Such  a  high  value  is  not  usually  practicable,  but  the 
value  of  eD  changes  but  little  when  S/V-  is  as  small  as  0.4.  From  0.37  to 
0.52  are  values  used  for  marine  turbines,  and  Peabody  gives  0.6  as  a 
common  ratio  of  S  to  V  for  electrical  work. 

The  peripheral  speed  of  marine  turbines  varies  from  110  to  210  ft.  per 
sec.  for  the  low-pressure  end,  and  from  70  to  130  ft.  for  the  high-pressure. 


THE  STEAM  TURBINE  261 

For  electrical  work,  the  low-pressure  speed  is  from  200  to  360  ft.  per 
sec.  for  the  low-pressure  and  100  to  135  ft.  for  the  high-pressure  end. 

As  already  stated,  blades  must  not  be  less  than  J^5  of  the  drum  diam- 
eter for  land  turbines,  but  are  sometimes  as  small  as  ^75  of  the  diameter 
for  marine  turbines,  at  a  sacrifice  of  efficiency.  They  should  not  be  longer 
than  J£  of  the  diameter  of  the  pitch  circle  for  the  best  results,  this  ap- 
plying at  the  low-pressure  end.  Speakman's  rule  is  that  the  blade  length 
shall  not  be  less  than  0.03  nor  greater  than  0.15  of  the  pitch  diameter. 

As  with  impulse  turbines,  the  square  of  the  mean  diameter  of  the 
blading  at  the  low-pressure  end  should  not  be  less  than  0.57  times  the 
output  in  kw.  for  a  28-in.  vacuum  according  to  Martin,  and  for  very 
high  vacuum  is  sometimes  made  equal  to  the  output  in  kw.  For  a 
double-flow  turbine  it  may  be  one-half  of  these  figures. 

Assuming  a  28-in.  vacuum  for  a  single-flow  turbine,  the  diameter  of 
pitch  circle  for  the  low-pressure  end  is  : 

D  ?  0.755VS  (62) 

which  may  be  determined  from  (39)  by  substitution. 

This  may  be  compared  with  (40).  It  may  usually  be  assumed  that 
the  only  variables  in  (40)  are  v  and  m;  letting  H  and  L  denote  high 
pressure  and  low  pressure  respectively, 

DL       SL      3    jmH    VL 

£-=s*==v^-^ 

In  practice  this  ratio  ranges  from  2  to  3.  To  keep  mH  and  mL  within  the 
prescribed  limits,  VL  will  be  less  than  the  specific  volume  corresponding 
to  the  vacuum  when  the  ratio  DL/DH  is  that  sometimes  found  in  practice. 
If  VL  is  taken  midway  of  the  portion  G,  Fig.  155,  the  results  will  more 
nearly  correspond;  in  this  case  the  exit  area  is  smaller  than  theoretically 
called  for  which  may  be  offset  by  the  use  of  wing  blades  or  semi-wing 
blades,  which  give  a  greater  exit  area,  and  the  result  may  be  checked  by 
(38).  The  pitch  diameter  of  the  intermediate  cylinder  may  be  made  a 
mean  proportional  between  the  high-  and  low-pressure  cylinders,  or: 

DI  =  VDHDL  (64) 

If  there  are  four  cylinders,  DA  and  DB  may  denote  the  first  and  second 
intermediate  respectively;  then: 

DA  = 
and, 

DB  = 


There  are  no  general  rules  for  proportioning  cylinders,  dividing  the 
work,  or  determining  the  number  of  barrels;  suggestions  are  given  by 


262  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

different  authorities,  but  they  are  not  in  agreement.  It  is  probable 
that  a  rather  wide  range  of  conditions  will  give  good  results  if  certain 
necessary  fundamentals  are  adhered  to. 

The  loss  due  to  tip  clearance  (see  Fig.  153)  may  be  considerable,  and  it 
is  obvious  that  this  is  greater  when  the  ratio  c/d  is  large.  Formulas 
for  allowable  tip  clearances  are  as  follows: 

Peabody c  =  0.00066D  +  0.01    1 

Martin c  =  QWIDD  +  0.005d  j 

where  c  is  in  inches;  DD  is  drum  diameter,  also  in  inches,  and  D  the 
diameter  of  pitch  circle  as  given  before. 

In  some  turbines  the  diameter  of  the  pitch  circle  is  constant  for  each 
cylinder,  but  it  is  more  usual  to  have  the  drum  diameter  constant,  with  a 
different  pitch  diameter  for  each  barrel.  A  compromise  may  be  made  in 
which  both  drum  and  pitch  diameter  change,  but  each  in  a  lesser  degree. 
For  a  constant  drum  diameter,  blade  height  and  pitch  diameter  are 
variables  and  must  be  determined  by  trial  and  error  for  great  accuracy; 
however,  this  seems  to  be  neglected  by  some  authorities,  the  pitch  circle 
being  assumed  constant.  The  heat  drop  is  proportional  to  V2,  which  is  in 
turn  proportional  to  D2.  Having  determined  blade  lengths  on  the 
assumption  of  a  constant  pitch  diameter  D,  if  n  is  the  calculated  number 
of  stages  in  the  barrel,  the  number  of  stages  may  sometimes  be  reduced  if 
there  are  a  large  number;  if  DA  is  the  actual  diameter  of  the  pitch  circle 
of  a  given  barrel  and  nA  the  new  number  of  blades;  then: 

nADA2  =  nD* 
or: 

nA  =  ^  (63) 

The  corrected  blade  length  dA,  which  with  actual  pitch  diameter  DA 
will  give  the  same  passage  areas  as  the  length  d  and  the  assumed  diameter 
Dis: 

DD*      D 


dA  --  ^Dd  ,     4         2 

The  drum  diameter  DD  (which  for  shrouded  blades  is  the  diameter  at 
inner  ends  of  blades)  may  be  found  for  any  barrel  on  a  cylinder,  D  and  d 
being  the  actual  pitch  diameter  and  blade  length  for  this  barrel.  The 
other  barrels  may  be  corrected.  It  is  obvious  that: 

DA  =  DD  +  dA. 

It  would  seem  that  the  blade  lengths  just  calculated  should  be  the  effect- 
ive length  d  —  c  of  Fig.  153.     Then  the  drum  diameter  Dc  used  in  cal- 


THE  STEAM  TURBINE 


263 


culation  would  be  the  actual  drum  diameter  plus  2c.  This  would  give 
the  worst  condition,  the  steam  leaking  over  the  tips  of  the  guides  not 
really  being  wholly  ineffectual. 

In  applying  (40)  and  (406),  it  may  be  seen  that  there  is  some  difficulty 
in  providing  ample  passage  for  the  steam  in  the  lower  stages  if  the  usual 
peripheral  velocity  limit  is  not  exceeded,  even  if  wing  blades  having  the 
maximum  practical  angle  are  employed.  This  is  especially  true  for  high 
rotative  speeds.  To  overcome  this,  the  last  two  or  three  rows  may  be 
increased  in  pitch  diameter  and  placed  upon  a  solid  disc,  which  will 
better  withstand  the  high  centrifugal  forces. 


f  W+JL   for  j  Blades 

A     ,ii 

!  \W+?    "  Larger  frlades 


FIG.  156. 

Peabody  gives  dimensions  and  proportions  of  blades  and  accessories 
as  recommended  by  Speakman,  and  these  are  shown  in  Fig.  156  and 
Table  43;  they  may  be  used  as  a  guide.  Tables  44  and  45  are  also  from 
the  same  source. 

Peabody  states  that  at  the  time  of  Speakman's  paper,  the  maximum 
speeds  of  Parsons  turbines  for  electrical  work  was  170  ft.  per  sec.  for 
high-pressure  blades,  and  375  ft.  for  low-pressure. 

Marine  turbine  rotative  speeds  are  governed  by  the  propeller  speed 
when  direct-connected,  a  compromise  being  necessary  between  the  most 
economical  speed  of  turbine  and  propeller.  The  compromise  is  unneces- 
sary when  driving  is  through  gears  or  electricity.  The  relative  merits  of 
gear  and  electric  drive  is  a  contested  subject,  especially  as  applied  to 
battleships. 

TABLE  43 


H... 
W.. 
P... 
C.. 


1" 


KG" 


K 


4" 
«' 

W 
KG" 


6" 


10' 


2H" 


W 


H" 


15* 


KG 


18" 
1* 


24' 


30" 
4" 


264  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  44. — (ELECTRICAL  WORK) 


Normal  output  kw. 

Peripheral  speed,  ft.  per  sec. 

Number  of 
stages 

R.p.m. 

First  expansion 

Last  expansion 

5000 

135 

330 

70 

750 

3500 

138 

280 

75 

1200 

2500 

125 

300 

84 

1360 

1500 

125 

.360 

72 

1500 

1000 

125 

250 

80 

1800 

750 

125 

260 

77 

2000 

500 

120 

285 

60 

3000 

250 

100 

210 

72 

3000 

75 

100 

200 

48                    4000 

TABLE  45. — (MARINE  WORK) 


Type  of  vessel 

Peripheral  speed, 
ft.  per  sec. 

8/V 

Num- 
ber of 
shafts 

H.p. 

L.p. 

High-speed  mail  steamers  

70-80 
80-90 
90-105 
85-100 
105-120 
110-130 

110-130 
110-135* 
120-150 
115-135 
130-135 
160-210 

0.45-0.50 
0.47-0.50 
0.37-0.47 
0.48-0.52 
0.47-0.50 
0.47-0.51 

4 
3  or  4 
3 
4 
3  or  4 
3  or  4 

Intermediate  steamers                       .        .  . 

Channel  steamers  

Battleships  and  large  cruisers            

Small  cruisers 

Torpedo  craft  

95.  Factors  Influencing  Turbine  Operation. — The  heat  factor  has 
been  used  in  the  preceding  discussion  of  turbine  design,  and  while  it  is 
more  reliable  to  use  values  determined  by  tests  than  to  rely  upon  labora- 
tory determinations  of  the  different  factors  upon  which  it  depends,  it 
is  well  to  briefly  notice  some  of  the  latter.  The  main  sources  of  loss  are : 

(a)  Nozzle  friction. 

(6)  Blade  friction. 

(c)  Disc  friction. 

(d)  Fan  effect  of  blades. 

(e)  Spilling  and  tip  leakage. 

(f)  Leakage  past  glands  and  diaphragm  packing. 

(g)  Degradation  of  heat  due  to  residual  velocity. 
(h)  Radiation. 

These  quantities  are  discussed  in  more  or  less  detail  in  the  more 
elaborate  treatises,  and  a  study  of  them  is  helpful  to  a  more  complete 


THE  STEAM  TURBINE 


265 


knowledge  of  turbine  operation.  Some  of  them  are  included  in  the  dia- 
gram efficiency  eD  already  discussed,  and  still  more  in  the  blade  efficiency 
eB  For  the  reaction  turbine,  Formula  (60)  gives  a  value  of  F  including 
most  of  these  factors,  and  it  is  clear  that  for  this  type  some  of  the  items 
do  not  apply. 

If  in  each  case,  one  minus  the  fraction  of  loss  be  the  efficiency  e,  the 
heat  factor  is  the  product  of  all  these  efficiencies ;  or : 

F  =  eAeBeceD,  etc. 

According  to  Martin,  the  heat  factor  depends  upon  a  certain  coefficient 
which  is  based  upon  construction  and  speed,  and  is: 


D\'(*L\* 

uoo/ 


(67) 


in  which  nB  is  the  number  of  rows  of  moving  blades.  The  relation  be- 
tween this  coefficient  and  the  heat  factor  is  given  by  a  curve  in  Martin's 
book  (The  Design  and  Construction .  of  Steam  Turbines),  but  may  be 
approximately  expressed  by  a  simple  formula,  thus : 

7 


If 


[f  7  =  from  30,000  to  120,000,     F  =  0.2 
:  from  120,000  to  300,000,     F 


100,000 


)  +  0.45 


a°4(ioorooo) 


(68) 
+  0.65    (69) 


FIG.  157. 

These  apply  especially  to  reaction  turbines,  but  may  be  used  for  other 
turbines  of  good  design.  It  is  assumed  that  all  turbines  having  the  same 
value  of  7  have  the  same  heat  factor;  this  may  be  affected  by  superheat 
or  other  factors  (correction  curves  are  given  by  Martin),  and  must  be 
used  with  judgment,  but  it  will  serve  as  a  check  upon  other  calculations 
or  assumptions. 

The  Condenser. — A  high  vacuum  is  of  much  greater  advantage  to  the 
steam  turbine  than  to  the  steam  engine;  this  may  be  explained  by  Fig. 
157,  which  is  a  pressure-volume  diagram  for  the  Rankine  cycle.  First 
assume  that  the  back-pressure  line  is  the  upper  boundary  of  the  shaded 


266  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

area.  For  the  turbine,  expansion  is  complete  and  is  from  a  to  c.  Com- 
plete expansion  is  not  practicable  for  the  steam  engine,  as  explained  in 
former  chapters,  so  ab  is  the  expansion  line.  Now  assume  an  increase  of 
vacuum,  bringing  the  back-pressure  line  down  to  the  bottom  of  the  shaded 
area.  The  gain  for  the  engine  is  the  shaded  area  up  to  6,  while  for  the 
turbine  with  complete  expansion  to  d,  the  gain  is  equal  to  the  entire 
shaded  area. 

As  about  double  the  amount  of  cooling  water  is  required  to  increase 
the  vacuum  from  26  to  28  in.  of  mercury,  the  cost  may  offset  the  gain  of 
power  for  the  steam  engine  in  some  cases;  the  gain  of  power  being  so 
much  greater  for  the  turbine,  the  higher  vacuum  is  of  decided  financial 
advantage,  in  spite  of  the  more  highly  efficient  condensing  apparatus 
required. 

Steam  passages  may  readily  be  determined  from  (11)  when  suitable 
steam  velocities  are  decided  upon.  This  reduces  to: 

vWH  ,7m 

a  -=  (70) 


With  an  initial  gage  pressure  of  150  Ib.  per  sq.  in.,  a  vacuum  of  28  in. 
and  a  heat  factor  of  0.6,  values  of  a,  found  by  substituting  the  value 
of  W  from  (3)  ,  are  nearly  as  follows  : 

17 

For  steam  inlet.  .  .  .  as  =  1.5  ^7-  (71) 

Vs 

TT 

For  exhaust  outlet  .  .  .  .  aE  =  150  y-  (72) 

To  allow  for  overload  and  sudden  fluctuation  of  load,  Vs  may  be 
made  75  ft.  per  sec.  This  is  some  lower  than  that  allowed  for  the  steam 
inlet  of  a  reciprocating  engine,  but  it  is  computed  upon  a  different  basis. 
The  heat  drop  per  Ib.  of  pressure  is  vastly  greater  at  the  vacuums  usually 
employed  than  at  initial  pressure;  the  velocity  of  flow  is  proportional  to 
the  square  root  of  the  heat  drop;  therefore  a  slight  drop  in  pressure  will 
cause  a  high  velocity  at  the  exhaust  pressure,  so  VE  may  be  much  higher 
than  Vs.  Assuming  VE  as  300  ft.  per  sec.,  (71)  and  (72)  become: 

•  -  .  «.-!  |*lj|     (73) 

and: 

a*  =  f  (74) 

Martin  gives  two  rules  for  the  area  of  the  exhaust  outlet,  the  first 
being  1  sq.  in.  for  each  25  Ib.  of  steam  passed  per  hour;  and  the  second, 
3  sq.  ft.  per  1000  horsepower  developed.  The  first  gives  VE  =  300 


THE  STEAM  TURBINE  267 

for  the  assumptions  already  made,  and  the  second  gives  VE  =  350, 
nearly.  It  is  good  practice  to  make  the  exhaust  passage  as  large  as  con- 
struction will  permit,  exceeding  the  area  given  by  (74)  if  possible. 

96.  Governing,  Rating  and  Overload.— The  speed  control  of  steam 
turbines  is  usually  accomplished  by  throttling,  although  in  some  cases 
the  flow  to  the  nozzles  is  controlled  by  a  number  of  valves,  a  part  of 
which  are  wide  open  and  the  remainder  closed,  depending  upon  load 
requirements.  The  latter  method  may  be  said  to  correspond  to  the 
automatic  cut-off  of  the  steam  engine.  In  both  methods,  large  overloads 
are  provided  for  by  admitting  high-pressure  steam  directly  to  some  lower 
stage;  this  is  also  under  governor  control. 

There  is  no  definite  standard  for  the  rating  of  steam  turbines.  They 
are  sometimes  rated  so  that  about  20  per  cent,  overload  may  be  carried 
without  opening  the  by-pass  to  a  lower  stage,  and  sometimes  the  rated 
load  is  the  maximum  load  obtainable  without  the  by-pass.  In  the  latter 
case  it  is  probable  that  some  allowance  is  still  made,  and  it  is  always  well 
to  have  the  nozzle  area  for  the  first  stage  a  little  in  excess  when  the  area 
is  controlled  by  a  number  of  valves. 

In  proportioning  the  nozzle  areas  for  the  second  and  lower  stages  of  a 
pressure-stage  impulse  turbine,  or  the  passages  between  blades  and  guides 
of  a  reaction  turbine,  the  weight  of  steam  must  be  known.  This  neces- 
sitates the  selection  of  a  load  for  which  these  areas  will  be  calculated. 
This  matters  less  with  the  reaction  turbine  on  account  of  the  small  pres- 
sure drop  per  stage,  but  a  decrease  from  the  load  for  which  the  nozzles  of  a 
pressure-stage  impulse  turbine  is  designed,  reduces  the  pressure  in  all 
but  the  last  stage,  which  may  result  in  the  last  wheel  running  nearly  or 
entirely  idle.  To  prevent  this,  the  last  stage  nozzles  may  be  reduced 
some  in  area.  It  is  obviously  not  wise  to  design  any  but  the  first-stage 
nozzles  for  the  maximum  load  (not  considering  the  by-pass),  and  this 
only  when  part  of  the  nozzles  are  cut  out  by  valves  at  lighter  loads,  either 
by  hand  or  by  the  action  of  the  governor. 

From  (11): 

V      144 

-  =  — w 
v         a 

or,  for  constant  area  as  in  most  turbines,  the  ratio  V/v  varies  directly  with 
the  weight  of  steam  passed  through.  The  determination  of  intermediate 
pressures  for  given  nozzle  areas  would  be  necessary  in  order  to  find  V,  but 
this  would  be  exceedingly  difficult  and  of  little  practical  value.  It  is 
apparent  that  both  V  and  v  change  with  change  of  load,  usually  in  oppo- 
site directions,  a*id  for  practically  constant  rotor  speed,  the  velocity 
diagram  would  be  altered.  Referring  to  Fig,  133,  the  reduction  of  V 


268  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

at  light  loads,  with  a  constant  or  if  anything,  slightly  increased  value 
of  S,  would  result  in  a  larger  value  of  the  angle  0.  For  this  reason  if  for 
no  other,  the  practice  of  making  the  entrance  angle  greater  than  0  as 
shown  in  Fig.  135,  seems  advisable. 

In  view  of  the  foregoing,  too  much  refinement  in  pressure  distribution 
may  be  ill-advised,  and  the  more  refined  method  of  Par.  90  may  perhaps  be 
superfluous.  It  must  also  be  remembered  that  all  such  calculations 
depend  upon  the  heat  factor,  which  is  not  known  with  accuracy. 

It  seems  rational  to  design  turbines  for  dry  saturated  steam ;  this  will 
provide  ample  capacity  when  superheated  steam  is  used.  For  high  degrees 
of  superheat,  however,  it  may  be  advisable  to  make  allowance  in  nozzle 
or  guide  blade  design.  Heat  distribution  may  be  determined  by  either 
the  simple  or  more  refined  method  of  Par.  90,  checked  as  in  columns  5  to 
7  of  Table  40,  and  any  desired  alterations  made. 

As  with  the  steam  engine,  the  increase  in  economy  due  to  superheating 
is  greater  than  that  theoretically  indicated,  while  the  gain  due  to  in- 
creased vacuum  is  less.  Very  liberal  exhaust  openings  increase  the  gain 
due  to  high  vacuum  and  should  be  provided  when  possible. 

Examples  of  the  application  of  the  foregoing  principles  to  design  are 
given  in  the  following  paragraphs.  These  are  given  to  direct  the  use  of 
the  equations,  and  it  is  not  claimed  that  the  results  coincide  with  actual 
practice  in  turbine  design.  In  what  has  preceded,  an  attempt  has  been 
made  to  bring  out  most  of  the  important  principles  relative  to  the  power 
of  a  turbine  in  a  general  way.  More  refinement  is  possible,  but  it  is 
doubtful  if  this  is  necessary  or  even  desirable.  The  practicability  of 
some  of  the  refinements  already  noted  must  be  left  to  the  designer. 

It  is  probable  that  in  certain  phases  of  design  considerable  latitude  is 
allowable,  but  there  is  no  doubt  about  the  truth  of  the  following  statement 
from  the  bulletin  of  the  General  Electric  Co.:  "In  order  to  obtain  the 
best  possible  efficiency  it  is  necessary  that  the  details  of  design  shall  be  in 
accordance  with  determinations  resulting  from  a  large  amount  of  experi- 
mental research,  and  the  experience  gained  in  a  very  extended  manu- 
facture." 

97.  Application  of  Formulas.  Impulse  Turbine. — The  design  of  a 
pressure-stage  turbine  with  two  velocity  stages  in  the  first  stage  involves 
practically  all  of  the  principles  of  the  impulse  turbine,  so  this  will  be 
taken  as  an  example. 

Problem:  Design  a  750-kw.  turbine  to  carry  a  steam  pressure  of  150 
Ib.  per  sq.  in.  gage  and  a  vacuum  of  28  in.  of  mercury.  The  speed  is  to 
be  3600  r.p.m.  and  the  steam  initially  dry  saturated.  Let  the  maximum 
velocity  at  the  pitch  line  of  blades  be  500  ft.  per  sec.  and  the  mechanical 


THE  STEAM  TURBINE  269 

efficiency  of    turbine  and  generator  combined  be  assumed  as  90  per 
cent. 

The  design  of  the  velocity  diagrams  is  first  required,  and  the  velocity 
coefficient  q  for  the  first  stage  will  be  taken  as  0.85  (see  Par.  84).  The 
nozzle  angle  a  will  be  taken  as  20  degrees  and  the  ratio  S/V  for  the  first 
stage  as: 

°^  =  0.1879. 
5 

The  diagram,  Fig.  158,  was  .drawn  to  scale  and  the  angles  chosen 
rather  arbitrarily,  and  as  results  were  fairly  good,  no  changes  were  made. 
Trigonometric  functions  were  found  by  measuring  the  diagram  and 
calculations  were  by  slide  rule.  The  functions  needed  for  drawing  blades 
and  for  calculation  are  as  follows : 

S      cos  on      0.9397 

=  =  — — —  =  — - —  =  0.1879,     cos  0i  =  cos  <x2  =  0.915,     0i  =  a2  = 

V  o  o 

23°-  48'. 


FIG.   158. 

According  to  Fig.  135,  0i  may  be  made  28  degrees  in  the  blade,  but 
this  does  not  change  the  velocity  diagram. 

cos  0i  =  0.924  0!  =  22°  -  29' 

cos  fa  =  cos  dl  =  0.868  fa  =  di  =  29°  -  46' 

cos  02  =  0.76  02  =  40°  —  32.     This  may  be  made  45 

degrees   in   the   blade.     As  velocities  have  not  been  definitely  decided 
upon,  measurements  in  inches  may  be  used;  then  (24)  gives: 

VWi  =  5.95 
and 

VW2  =  1.0905. 
The  ratio  of  work  is: 

V^l-    AM_    -545 
VW2       1.0905  " 

instead  of  3,  as  given  in  Par.  88  for  frictionless  flow  with  symmetrical 
blades, 


270  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

From  (30),  the  diagram  efficiency  is: 

2  X  0.8  X  7.855 
~4L2CT 

Fig.  158  is  such  a  diagram  as  Fig.  140,  but  drawn  with  the  apexes  of  all 
triangles  at  the  same  point,  a  convenient  construction  for  design. 

The  velocity  diagram  for  all  other  stages  is  shown  in  Fig.  159,  drawn 
in  the  same  way,  in  which  q  is  taken  as  0.75  (Rateau's  value).  Other 
quantities  are: 

a  =  20,        |L«  0.5  cos  a  =  0.47,  and  K  in  (27)  is  0.44. 

cos  $  =  0.885         0  =  27°  -  45',         cos  0  =  0.806,     0  =  36°  -  18'. 

This  last  may  be  made  40  degrees  in  the  blades  according  to  Fig.  135. 
Then  using  inches  as  before,  from  (27) : 

eD=  0.44[(0.75  X  1.096)  +  1]  =  0.783. 


FIG.  159. 

From  Par.  95,  assuming  the  loss  by  nozzle  and  disc  friction,  gland 
leakage  and  radiation  combined  to  be  20  per  cent.,  the  heat  factor  for  the 
first  stage  is: 

F  =  0.80  X0.69=  0.552. 
For  the  other  stages: 

F  =  0.80  X  0.783  =  0.626. 

Assume  tentatively  that  the  blade  pitch  line  of  all  wheels  travels  500 
ft.  per  sec.;  then  for  the  first  stage: 

2660'  •- 


From  (37),  neglecting  FR  and  y,  the  heat  drop  for  this  stage  is: 
.  2660  2 


Peabody's  entropy  table  will  be  used  for  heat  quantities,  volumes,  etc., 
in  these  calculations;  the  nearest  tabular  value  of  absolute  pressure  corre- 
sponding to  150  lb.  gage  is  164.8,  and  to  28-in.  vacuum,  1.005,  and  these 


THE  STEAM  TURBINE  271 

will  be  used  for  PI  and  P2.  The  heat  content  Ci  is  1193.3,  and  C2  is 
871.1  B.t.u.  The  available  heat  at  constant  entropy,  which  for  initially 
dry  saturated  steam  is  1.56,  is: 

Ci  -  C2  =  322.2  B.t.u. 
The  heat  left  for  the  remaining  stages  is  : 

322.2  -  141  =  181.2. 
For  these  stages: 

' 


From  (37),  neglecting  FK  and  y  as  before,  the  heat  drop  per  stage  is: 

1063*  . 

"(2237?" 

The  number  of  these  stages  after  the  first  stage  will  be  : 

181.2    ' 
"227  =  8' 

As  PR  and  y  were  neglected,  the  velocity  may  exceed  the  limiting 
velocity  of  500  ft.  per  sec.  if  the  tentative  values  are  adopted.  Let  us  now 
arbitrarily  take  the  number  of  single  wheel  stages  to  be  10,  making  11 
stages,  and  12  rows  of  moving  blades.  Assume  a  heat  drop  per  stage 
along  constant  entropy  of  20  B.t.u.  The  heat  drop  for  the  first  stage  is 
then: 

Ac  =  322.2  -  200  =122.2. 

Assume  the  heat  factor  to  be  a  compromise  between  those  found  from 
the  efficiencies  of  the  two  diagrams,  as  follows: 


This  is  perhaps  rather  low,  but  conservative.     From  Table  41,  R  =  1.08. 
The  reheat  factor  for  the  first  stage,  from  (36),  is: 


For  the  other  stages: 


Taking  the  friction  factor  for  all  nozzles,  y  =  0.1,  (37)  gives  for  the 
first  stage: 

V  =  223.7  V0.9  X  1.05  X  122.2  =  2405. 

Also: 

S  =  0.1879  X  2405  =  452     and     D  =  60X01ft2J*452  -  28.75  in. 

ODUU 


272  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

For  the  other  stages: 

V  =  223.7VO^~XT.075^T20  =  983. 
Also: 

S  =  0.47  X  983  =  462.5  and  D  =  6°  X  *  *  462'5  =  29.375  in. 


A  check  calculation  may  now  be  made  by  means  of  (40)  and  (40a); 
from  the  latter,  at  the  condenser  pressure,  1.005  lb.: 

Cx  =  871.1  +  (0.4  X  322.2)  -  (0.442  X  1.075  X  20)  =  990.48. 
Interpolating  in  Peabody's  entropy  table  for  the  corresponding  volume 
gives : 

v  =  295. 
Then,  assuming  m  as  0.2,  and  k  as  0.9,  (40)  gives: 

n=oi*3/-  750  X  295  X  0.47 

\0.9  X  0.2  X  0.9  X  3600  X  0.6  X  322.2  X  0.342 

Properly,  m  should  be  multiplied  by  the  ratio  di/dz  as  explained  in 
connection  with  equations  (38)  and  (39),  but  for  the  sake  of  variety  in 
the  problem  this  will  be  neglected  at  this  place  and  a  correction  made  for 
the  last  stage.  It  may  then  be  assumed  that  the  diameters  already  found 
are  correct. 

In  finding  the  nozzle  areas,  the  steam  weight  must  be  known.  Calcu- 
lations will  be  made  for  rated  load  at  maximum  steam  pressure,  a  condi- 
tion probably  not  obtaining  in  practice  except  when  governing  is  effected 
by  cutting  out  valves,  which  will  be  assumed  to  be  this  case ;  then  enough 
extra  nozzles  must  be  provided  for  the  first  stage  to  take  care  of  the  over- 
load. For  governing  by  throttling,  the  nozzles  of  the  higher  stages  should 
be  designed  for  overload  at  maximum  steam  pressure. 

From  (5),  the  turbine  horsepower  is: 

1.34  X  750 


~09 
From  (3),  the  water  rate  is: 


_ 


and:     - 

WH         13.15  X  1118      .  nQ 

=  60X60  =       ~~3600~ 
From  (11),  the  total  nozzle  area  is: 

a  =  144  X  4.08  |P  =  603^- 

For  the  first  stage,  V  =  2405,  and  a  =  0.2503v. 
For  the  other  stages,  V  =  983,  and  a  =  0.614v, 


THE  STEAM  TURBINE 


273 


TABLE  46 


Stage  no. 

2 

Cn  +  (l-Fs)FR-A.nC- 
FR-An  +  iC 

3 
P 

4 

<£ 

5 

V 

6 

a 

Steam 
chest 

1193.30 
128.31 

164.800 

1.56000 

2.750 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

1064  .  99 
54.99 

30.125 
22.315 
16.375 
11.923 
8.633 
6.178 
4.374 
3.069 
2.138 
1.470 
1.016 

1  .  63737 
1.65112 
1  .  66507 
1  .  67924 
1  .  69391 
1  .  70899 
1  .  72447 
1  .  74020 
1.75629 

1.77141 
At  P  =  1  .  005 

1  .  78943 

12.260 
16.999 
22  .  570 
30.207 
40.513 
55.077 
75.805 
104  .945 
146.515 
207.032 
291.930 

7.53 
10.42 
13.82 
18.55 
24.90 
33.80 
46.50 
64.30 
90.00 
127.20 
175.00 

1119.98 
21.50 

1098.48 
9.50 

1107.98 
21.50 

1086.48 
9.50 

1095.98 
21.50 

1074.48 
9.50 

1083.98 
21.50 

1062.48 
9.50 

1071.98 
21.50 

1050.48 
9.50 

1059.98 
21.50 

1038.48 
9.50 

1047.98 
21.50 

1026.48 
9.50 

1035.98 
21.50 

1014.48 
9.50 

1023.98 
21.50 

1002.48 
9.50 

1011.98 
21.50 

990.48 
9.50 

999.98 

18 


274 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  method  of  pressure  distribution  of  Table  41,  Par.  90  is  used,  the 
results  being  given  in  Table  46.  Considerable  accuracy  is  required  due  to 
the  small  pressure  drops  in  the  later  stages,  and  cross  interpolation  is 
resorted  to  in  the  use  of  the  entropy  table.  Careful  use  of  the  slide 
rule  is  permissible  in  making  interpolations.  As  stated  in  Par.  90,  the 
check  on  this  system  is  the  last  pressure,  which  in  this  case  is  1.016  in- 
stead of  1.005,  a  difference  which  has  no  practical  bearing. 

Total  nozzle  areas  are  given  in  Col.  6.  This  is  the  exit  area  at  rated 
load  for  the  first-stage  nozzles.  In  all  but  the  first  stage  the  pressure 
is  greater  than  58  per  cent,  of  that  in  the  stage  above  it,  so  that  convergent 
nozzles  are  used.  In  the  first  stage  divergent  nozzles  are  required. 

The  pressure  in  the  throat  of  the  first-stage  nozzle  is : 

164.8  X  0.58  =  95.5. 

The  heat  content  at  entropy  1.56  and  this  pressure  is  1149.29,  and  the  cor- 
responding volume  4.44.  The  heat  drop  to  the  throat  is  44.01.  Assum- 
ing the  friction  factor  for  flow  to  the  throat  as  0.02,  the  velocity  at  the 

throat  is:  

V  =  223.7 -v/0.98  X  44.01  =  1470  ft.  per  sec. 

There  is,  of  course,  no  heat  factor  at  this  point.  The  throat  area  for 
rated  load  is  then: 

v       603  X  4.44 


a  =  603  ~  = 


1470 


1.823  sq.  in. 


y//// 


FIG.  160. 

The  first  stage  nozzles  are  shown  in  Fig.  160:  these  are  machined,  out  of 
bronze  and  bolted  to  the  diaphragm  as  shown  in  Fig.  460,  Chap.  XXXIII. 
Fig.  161  shows  second-  and  third-stage  nozzles  machined  in  the  diaphragm, 
while  Fig.  162  shows  cast-in  nozzles  for  the  remainder  of  the  stages.  The 
first-stage  nozzles  have  partial  admission  and  are  located  on  one  side  of 
the  diaphragm.  The  second-  and  third-stage  nozzles  have  partial  admis- 
sion but  extend  over  a  larger  portion  of  the  periphery;  they  are  equally 
divided  between  the  two  halves  of  the  diaphragm.  The  cast-in  nozzles  of 
the  remaining  stages  have  full  admission. 


THE  STEAM  TURBINE 


275 


The  number  of  nozzles  have  been  assumed,  keeping  in  mind  the  blade 
length  and  are  given  in  Table  47.     As  computed  in  Table  46  there  would 


FIG.  161. 


be  15  first-stage  nozzles  of  the  diameter  given;  20  per  cent,  overload  is 
provided  for  making  18  first-stage  nozzles,  and  this  number  appears  in 
Table  47. 


FIG.  162. 
TABLE  47 


Stage 

Number  of  nozzles 

^     \.    d,in. 

Fraction  of  circum- 
ference 

1-throat 

18 

0.394 

0.497 

1-exit 

18 

0.800 

0.497 

2 

20 

0.832 

0.846 

3 

20 

0.940 

Full 

4 

56 

0.650 

Full 

5 

56 

0.880 

Full 

6 

56 

1.200 

Full 

7 

56 

1.640 

Full 

8 

56 

2.275 

Full 

9 

56 

3.180 

Full 

10 

56 

4.150 

Full 

11 

56 

6.320 

Full 

276 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


For  the  conditions  assumed,  the  blading  data  are  given  in  Table  48. 
For  the  first  stage,  the  entrance  height  of  the  first  row  of  blades  is  made 
about  6  per  cent,  greater  than  the  diameter  of  nozzle  exit.  The  entrance 
to  the  guides  and  second  row  of  blades  equals  the  preceding  exit  heights, 
which  are  found  in  each  case  by  (19),  neglecting  the  drying.  The  ratios 
sin  A  /sin  B  are  given  in  Table  48. 

TABLE  48 


A 

igle 

a, 

in. 

Sin  A 

Entrance 

Exit 

Entrance 

Exit 

Sin  B 

1st 

167 

28°-0' 

22°-29' 

0.850 

0.900 

.055 

1 

Fixed 

84 

29°-46' 

23°-48' 

1.050 

1.110 

.230 

2d 

156 

45°-0' 

29°-46' 

1.260 

1.450 

.305 

2 

162 

40°-0' 

27°-45' 

0  920 

1  060 

270 

3 

162 

40°-0' 

27°-45' 

1.030 

1.200 

.270 

4 

162 

40°-0' 

27°-45' 

0.715 

0.825 

.270 

5 

162 

40°-0/ 

27°-45' 

0  970 

1  120 

270 

6 

162 

40°-0' 

27°-45/ 

1  320 

1.520 

.270 

7 

162 

40°-0' 

27°-45' 

1.700 

2.080 

.270 

8 

162 

40°-0' 

27°-45' 

2  500 

2  875 

270 

9 

162 

40°-0' 

27°-45' 

3.500 

4.050 

.270 

10 

162 

40°-0' 

27°-45' 

4.570 

5.280 

.270 

11 

162 

40°-0' 

27°-45' 

7  000 

8  050 

270 

After  drawing  the  blading  in  this  way  as  shown  in  Fig.  163  by  the  full 
lines,  straight  lines  are  drawn  from  nozzle  extremities  to  the  last  blade 


FIG.  163. 


extremities  as  mentioned  in  the  latter  part  of  Par.  88.     Then  the  entrance 
height  of  guides  and  blades  are  brought  up  to  the  height  given  by  the 


THE  STEAM  TURBINE 


277 


dotted  straight  line.  This  change  is  arbitrary  and  is  shown  in  heavy 
dotted  lines.  The  completed  arrangement  showing  shrouding,  etc.,  is 
shown  in  Fig.  164.  The  sections  of  blading  with  their  trajectory  curves 
are  shown  in  Figs.  149  and  150. 


FIG.  164. 


Returning  to  the  last  stage,  the  blade  length  given  in  Table  48  is 
greater  than  D/5,  which  has  been  assumed  as  a  limit.  Correcting  m 
as  mentioned  in  connection  with  (19),  gives: 


278  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Substituting  this  in  (39)  gives: 

D  -  fi  77    I-  4.08  X  295  ~  _        . 

\0.9  X  0.157  X  983  X  0.342 

This  assumes  the  same  heat  drop  and  nozzle  angle  as  before.     Then: 

irDN 


s 


720 


534 


and 


The  nozzle  height  is  now: 

d  =  mD  =  0.157  X  34  =  5.32 

and  this   checks   with   the   area   in   Table  46.     The  maximum  blade 
height  is4 

d2  =  l.27d  =  6.75  in. 

The  minimum  may  be  made : 

di  =  1.12  X  5.32  =  6  in. 


FIG.  166. 

The  blade  height  is  now  J£  the  pitch  diameter. 

The  change  of  S/V  modifies  the  form  of  the  velocity  diagram  and  the 
blade  section.  The  velocity  diagram  is  shown  in  Fig.  165  and  the  sec- 
tions with  trajectory  in  Fig.  166.  The  calculations  are: 

cos  6  =  0.778 
and 

6  =  38°-55'. 


THE  STEAM  TURBINE 

Make  the  angle  45  degrees. 

sin  38°-55'  =  0.628. 
For  the  same  ratio  as  the  other  stages: 

0.628 


279 


sin  0  = 


1.27 


=  0.494 


and 


0  =  29°-37' 
cos  0  =  0.869. 


Then  from  (27),  eD  =  0.79; 

which  is  greater  than  for  the  other  stages  due  to  the  greater  ratio  of 
cos  0/cos  6. 

The  number  of  blades  on  the  last  wheel  will  be  176,  and  their  form  is 
almost  identical  as  for  those  on  the  second  wheel  of  the  first  stage;  the 


FIG.  167. 

trajectory,  however,  is  quite  different,  due  to  the  difference  in  the  ratio 
of  relative  velocity  of  steam  to  blade  speed. 

Reaction  Turbine. — Using  the  same  power,  rotative  speed  and  pres- 
sures as  for  the  impulse  turbine,  the  dimensions  of  a  reaction  turbine  will 
be  determined.  The  total  heat  factor  will  be  the  same  and  the  steam 
assumed  to  be  dry  saturated  initially. 

Let  it  further  be  assumed  that: 


a  =  18  degrees,     S/V  =  0.6, 


0.89,    j  =  0.5     eM=  0.9 


and  let  the  velocity  diagram  be  as  in  Fig.  167. 

The  diagram  efficiency  from  (58)  is  0.78.  As  preliminary  assump- 
tions let  SH  =  150  and  SL  =  360,  the  subscripts  H,  I  and  L  denoting 
high-,  intermediate-  and  low-pressure  cylinders  respectively.  Then: 


150 
0.6 


=  250 


280  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and 


From  (54)  and  (55a),  neglecting  FR,  the  heat  drop  per  stage  is: 

A//C  =  1.25 
and 

ALC  =  7.2. 

Taking  Fs  =  F  and  neglecting  FR,  the  heat  content  at  exit  from  the 
last  row  of  guides  is,  from  (406),  997.1  and  v  =  298.  Assuming  k  =  1  for 
all  stages,  and  ra  =  0.2  for  the  last  stage,  (40)  gives,  taking  a  =  45  for 
wing  blades: 

DL  =  24.7  in. 

Then  SL  =  387.5,  which  is  excessive,  for  the  assumptions  made. 

With  SH  =   150,  DH   =   9.5  in. 

As  w  =  4.08,  the  heat  drop  A#C  =  1.25,  and  va  =  2.8.  Then  (38) 
gives:  ra  =  0.075.  As  this  is  large,  assume  DH  to  be  10  in.  Then 
restricting  SL  to  360,  DL  =  23  in.  From  (64),  />/  =  15.25  in.  Then: 

Sa  =  157  Sz  =  239  SL  =  361 

VH  =  278  7,  =398  VL  =  602 

Neglecting  wing  blades  on  the  low-pressure  cylinder,  the  heat  drop 
oer  row  of  moving  blades  (or  per  stage)  is: 

A#C  =  3.08,  A7C  =  6.34,  ALC  =  14.5. 

It  now  remains  to  distribute  the  work  among  the  three  cylinders. 
This  is  rather  arbitrary,  but  we  will  take  three  barrels  per  cylinder  and 
designate  the  barrels  by  subscripts  1,  2,  3,  etc. 

By  taking  DH  =  10.75  and  re-arranging  the  data,  the  following  results 
are  obtained,  in.  which  na ,  nI}  and  nL  denote  the  number  of  rows  of  moving 
blades,  or  the  number  of  stages  on  the  respective  cylinders,  and  n\,  n2,  etc. 
the  stages  per  barrel. 

DL=    23" 

SL     =  361 
'VL   =   602 

nL  =  10 
AL<?  =    14.5 
tttf-AtfC  =  63.2  W/-A/C  =114  nL-&LC  =  145 


DH  =  10.75" 

Dj  =    15.25" 

Sa  =  168.6 

St    =  239 

Va   =   281 

Vf     =  389 

nH  =  20 

HI  =   18 

W?  =  3.16 

A/C  =  6.33 

\HC  =  63.2 

n/'A/C  =   114 

The  number  of  rows  per  barrel  and  the  heat  drop  per  barrel  are  as 
follows : 


THE  STEAM  TURBINE 


281 


Cyl.  No.  1 
Cyl.  No.  2 
Cyl.  No.  3 


7 
7 
6 
6 
6 
6 
4 
4 
_2 
48 


A2C 
A3C 

= 

22. 
22. 
18. 

12 
12 
96 

A4C 
A5C 

A6C 

= 

38. 
38. 
38. 

00 
00 
00 

A7C 

A8C 
A9C 

pr 

58. 
58. 
29. 

00 
00 
00 

322.20 


These  may  be  arranged  upon  a  distribution  diagram  similar  to  Fig.  155, 
and  this  is  done  in  Fig.  168.  Then  heat  drops  to  the  center  of  each  barrel 
are  taken  and  denoted  by  A,  B,  C,  etc. 
The  distribution  factors  are  then  scaled 
from  the  diagram  and  Table  49  com- 
puted after  the  manner  of  Table  40. 
Nozzle  areas  and  blade  lengths  are  added 
to  the  table.  The  latter,  computed  for 
the  center  of  the  barrel,  applies  to  all 
blades  and  guides  for  that  barrel.  It 
will  be  found  that  blade  lengths  for  the 
two  last  barrels  are  excessive,  and  to  ac- 
commodate the  steam  they  must  either 
be  placed  upon  a  larger  wheel,  or  wing 
blades  (or  possibly  semi-wing)  must  be 
provided.  If  m  =  0.2,  the  last  two 
barrels  will  have  blades  4%  in.  long. 
Then  from  (13),  for  the  eighth  barrel: 

sin  a  =  0.326  FIG.  168. 

and 

a  =  19  degrees. 

This  is  but  slightly  greater  than  the  other  blade  angles.  For  the  last 
stage : 

sin  a  =  0.682 
and 

a  =  43  degrees 

which  is  for  a  wing  blade.  As  this  was  computed  from  the  center  of  the 
last  heat  drop,  it  would  not  give  full  area  at  condenser  pressure,  but  as 
there  are  but  two  rows  on  the  last  barrel,  the  difference  may  be  neglected. 


282 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


It  must  be  remembered  that  the  same  discrepancy  is  found  in  every 
barrel,  the  blades  not  being  lengthened  as  theory  indicates  that  they 
should. 

TABLE  49 


1 

Stage 

2 

FD-AC 

3 

4 
P 
(Absolute) 

5 

6 

7 

V 

8 
a 

9 
a,  in. 

Steam 
chest 

1193.30 
11.90 

164.800 

1193.30 
6.64 

1.56 

.... 

A 

11.90 

1181.40 

143  .  500 

1186.66 

1.57 

3.109 

6.11 

0.55 

23.60 

13.28 

B 

23.60 

1157.80 

105.800 

1173.38 

1.58 

4.098 

8.04 

0.72 

21.75 

12.33 

C 

21.75 

1136.05 

81.100 

1161.05 

1.59 

5.218 

10.90 

1.00 

29.80 

17.10 

D 

29.80 

1106.25 

54.100 

1143.95 

1.61 

7.508 

11.40 

0.71 

39.10 

22.80 

E 

39.10 

1067.15 

31.420 

1121.15 

1.64 

12.310 

18.70 

1.27 

38.38 

•22.80 

F 

38.38 

1028.77 

17.520 

1098.35 

1.66 

21.090 

32.00 

2.16 

48.00 

22.80 

G 

48.00 

980.77 

8.020 

1069  .  55 

1.70 

42  .  670 

41.50 

1'.86 

55.75 

34.80 

H 

55.75 

923.02 

2.812 

1034.75 

1.75 

111.700 

109.00 

4.96* 

40.75 

26.10 

I 

40.75 

882  .  27 

1.271 

1008.65 

1.78 

235.000 

228.00 

10.20* 

*  Make  4.625  in.,  the  last  stage  being  wing  blades. 

Blade  widths  for  the  various  lengths  may  be  selected  from  Table  43. 

In  neither  of  these  examples  is  it  claimed  that  conditions  are  met  in  the 
best  manner.  The  heat  factor  chosen  is  lower  than  should  be  found  in 
such  machines,  but  it  is  probable  that  it  is  more  nearly  the  usual  value  for 
units  of  this  size. 

For  each  example,  the  steam  inlet  from  (73)  is  21.3  sq.  in.  and  the 
diameter  5.22  in.  The  next  larger  standard  pipe  size  is  6  in.  The  ex- 
haust opening  is  from  (74),  559  sq.  in.,  which  if  circular  would  be  26.7  in., 
or  in  a  standard  pipe  size,  28  in.  A  certain  Curtis  turbine  of  the  same 
power  has  an  exhaust  connection  30  in.  in  diameter. 


References 

The  design  and  construction  of  steam  turbines  — 

The  steam  turbine 

The  theory  of  the  steam  turbine 

Westinghouse  45,000-kw.  cross-compound  turbine , 


H.  M.  Martin. 
C.  H.  Peabody. 
Alexander  Jude. 
Power,  April  18,  1916. 


PART  V— MECHANICS 

CHAPTER  XVI 
THE  SLIDER  CRANK 

98.  Introduction. — The  slider-crank  mechanism  is  the  transmission 
machinery  of  the  reciprocating  engine.  With  certain  assumptions  it 
permits  of  a  very  rigid  mathematical  analysis,  involving  trigonometry 
and  the  trigonometric  functions  of  the  calculus,  subjects  with  which  many 
good  designers  are  not  at  all  intimate.  No  designer  of  reciprocating 
engines  is  well  equipped  who  cannot  deal  in  a  fairly  thorough  manner 
with  problems  involving  the  slider  crank,  so  in  this  chapter  an  attempt  is 
made  to  give  a  treatment  which  will  cover  essential  features  in  as  simple 
manner  as  possible. 

For  a  thorough  study  of  the  movement  of  the  connecting  rod  and  the 
forces  resulting  therefrom,  the  reader  is  referred  to  the  excellent  work 
of  Prof.  W.  E.  Dalby,  The  Balancing  of  Engines,  in  which  derivations  of 
formulas  for  acceleration  used  in  this  book  are  given. 

It  is  believed  that  the  treatment  will  be  of  greater  practical  use  if  the 
convention  respecting  signs  of  trigonometric  functions  is  ignored.  The 
formulas  will  therefore  give  absolute  values  and  the  proper  signs  for 
plotting,  found  by  inspection,  will  be  designated  for  the  different  quad- 
rants, usually  on  a  diagram  employed,  and  in  tables. 

An  attempt  to  account  for  friction  is  apt  to  involve  more  error  than 
its  neglect,  therefore  it  is  ignored. 

To  simplify  expression,  all  dimensions  are  in  feet  in  this  chapter. 

Notation. 

w  =  weight  in  pounds  of  section  of  connecting  rod.  Also  weight  in 
general. 

W  =  weight  in  pounds  of  any  part  or  collection  of  parts. 

P  =  force  due  to  steam  or  gas  pressure.  To  this  may  be  added— 
if  desired — inertia  of  the  reciprocating  parts  and  sometimes, 
for  approximate  work,  the  inertia  of  the  connecting  rod;  for 
vertical  engines  the  weight  of  the  reciprocating  parts  is  added 
on  the  down  stroke  and  subtracted  on  the  return  stroke. 

283 


284  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

G  =  force  in  pounds  due  to  gravity,  exerted  by  connecting  rod. 

C  =  force  exerted  by  gravity  of  crank  and  pin;  this  may  include 
counterbalance,  in  which  case  it  may  have  a  negative  value. 

F  —  force  due  to  acceleration,  in  pounds. 

T  =  total  turning  effort  in  pounds  at  crank  circle,  due  to  combined 
forces. 

L  =  length  of  connecting  rod  in  feet. 
L0  =  distance  in  feet  from  center  of  crosshead  pin  to  center  of  gravity 

of  connecting  rod. 
LP  =  same  to  center  of  percussion  of  rod. 

I  =  any  measurement  in  feet. 

R  =  radius  of  crank  circle  in  feet. 

n  =  L/R. 

rc  =  distance  in  feet  from  center  of  shaft  to  center  of  gravity  of 
crank,  pin,  or  counterbalance. 

r  =  radius  of  gyration  in  feet,  of  connecting  rod  about  crosshead  pin 
center. 

7  =  moment  of  inertia  of  rod  in  pound-feet  about  center  of  crosshead 
pin. 

x  =  any  measurement  in  feet  parallel  to  line  of  stroke;  also  specific 

measurements  in  this  direction  as  given  on  diagrams. 
y    =  same  normal  to  line  of  stroke. 

e  =  efficiency. 

V  =  velocity  in  feet  per  second. 

w  =  angular  velocity  in  radians  per  second. 
N  =  r.p.m. 

S  =  piston  speed  in  feet  per  minute. 

a  =  acceleration  in  feet  per  second  per  second. 

g  =  acceleration  due  to  gravity  (=  32.16). 

TT  =  3.1416. 

The  following  subscripts  are  generally  used : 

x  and  y  refer  to  quantities  in  direction  of  line  of  stroke  and  normal  to 
it. 

A    pertains  to  cylinder  end  of  engine. 

B    pertains  to  shaft  end  of  engine. 

H  refers  to  head  end  of  stroke. 

C    refers  to  crank  end  of  stroke. 

T  signifies  the  turning  effect  of  a  force. 

N  refers  to  the  normal  component. 

n    refers  to  any  number. 

L    pertains  to  force  along  connecting  rod. 


THE  SLIDER  CRANK 


285 


R    pertains  to  force  along  crank. 

W  refers  to  effect  of  weight  only. 

P    pertains  to  force  along  piston  path. 

99.  Properties  of  the  Connecting  Rod. — If  a  finished  rod  is  available, 
the  center  of  gravity  may  be  found  by  weighing  each  end.  Referring 
to  Fig.  169: 


h - -L 

k L* H 


T 


FIG.  169. 


From  which: 


(1) 


The  center  of  percussion  may  be  found  by  hanging  the  rod  upon  a 
knife  edge  at  the  center  of  the  crosshead  pin  bearing  as  shown  in  Fig. 
170;  support  a  plumb  line  from  a  point  near  by.  Adjust  the  plumb  bob 
until  it  swings  in  unison  with  the  rod,  and  the  length  of  the  line  to  the 
center  of  the  bob  measures  the  distance 
from  the  center  of  the  crosshead  pin  to  the 
center  of  percussion  of  the  rod. 

The  radius  of  gyration  is  a  mean  pro- 
portional between  these  two  distances  or: 
r  =  VLGLP  (2) 

If  no  rod  is  available  the  weight  must 
be  estimated  from  the  drawing.  The  rod 
may  be  divided  into  sections  as  shown  in 
Fig.  171.  The  weight  of  each  section  is  esti- 
mated and  designated  by  a  subscript. 
From  any  line,  as  A  B,  measure  the  dis- 
tance to  the  center  of  gravity  of  each  sec-  FlG  170 
tion ;  then  the  following  equation  holds : 

W(h  +  LG)  =  Will  +  ^2  +  Wa  etc. 
From  which: 

LG  =  ^  -  h  (3) 


The  stub  ends  may  be  divided  into  as  many  sections  as  desirable,  or 


286 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


I  may  be  taken  to  the  approximate  center  of  the  complete  stub.  For 
certain  forms  of  rod  the  center  of  the  body  may  be  found  more  easily, 
but  the  method  given  provides  data  in  a  usable  form  for  what  follows. 

By  a  similar  approximate  method  the  moment  of  inertia  may  be 
determined  directly.  Taking  moments  about  the  center  of  the  crosshead 
pin  instead  of  the  line  AB  gives: 

As  I  =  Wr\ 

IW 
I 


r  = 


(5) 


The  center  of  percussion  is,  from  (2): 


i<-—  /     jift /  __.„_____.__.• 

(r^\\ — s — i — r~ 


(6) 


FIG.  171. 

Theoretically,  the  greater  the  number  of  sections  into  which  the  rod 
is  divided,  the  greater  the  accuracy,  but  this  may  be  carried  too  far  for 
accurate  practical  work;  it  will  probably  suffice  if  the  length  of  the  section 
is  about  equal  to  the  mean  depth  or  diameter. 

For  the  crosshead  stub,  the  portion  on  either  side  of  the  pin  center 
must  be  taken  separately  and  not  considered  as  the  entire  stub  weight 
concentrated  at  the  center  of  gravity;  parts  on  both  sides  add  to  the 
inertia  and  do  not  neutralize  each  other  as  in  the  case  of  gravity.  Practic- 
ally the  inertia  effect  is  relatively  small  near  the  crosshead  pin  and  too 
much  refinement  is  uncalled  for.  For  this  reason  it  is  not  necessary  to 
add  the  moment  of  inertia  of  each  section  to  wl*,  although  this  is  theore- 
tically correct. 

Turned  taper  rods  are  often  used  and  in  some  cases  the  maximum 
diameter  is  at  the  center.  The  rod  may  then  be  divided  into  parts  com- 
posed of  the  stub  ends  and  one  or  two  frustums  of  a  cone.  The  volume, 
center  of  gravity,  and  moment  of  inertia  of  the  frustum  of  a  cone  may 
be  found  by  the  following  formulas,  in  which  di,  dz  and  dQ  are  diameters 


THE  SLIDER  CRANK 


287 


in  ft.  at  the  small  and  large  ends  and  at  the  center  of  gravity  respectively, 
wc  the  weight  per  cu.  ft.  and  v  the  volume  in  cu.  ft.  Let  l\  be  the  dis- 
tance from  the  small  end  to  the  center  of  gravity,  1%  the  same  to  the  large 
end,  and  I  the  sum  of  the  two  as  shown  in  Fig.  172.  The  volume  is: 

The  weight  is: 

W  =  wcv. 

The  distance  from  the  small  end  to  the  center  of  gravity  is : 


± 


FIG.   172. 

The  moment  of  inertia  about  the  center  of  gravity  is: 

in  which: 

do  =  c 


If  1G  is  the  distance  from  the  center  of  gravity  to  the  center  of  the 
crosshead  pin,  the  moment  of  inertia  about  the  latter  is : 

IA  =  I  +  W10*. 

If  the  rod  has  the  largest  diameter  at  or  near  the  center,  forming  two 
frustums,  they  may  both  be  treated  in  this  manner.  The  product  of 
the  weight  of  the  crank-end  stub  and  the  distance  of  its  center  of  gravity 
from  the  crosshead  pin  center,  added  to  the  values  of  I  for  the  two  parts 
of  the  rod  (the  two  frustums),  and  the  moment  of  inertia  of  the  crosshead 
stub — which  is  relatively  small — give  the  total  moment  of  inertia  for 
this  form  of  rod  more  easily  and  with  greater  accuracy  than  the  method 
of  Formula  (4). 

100.  Piston  Velocity. — In  Fig.  173,  the  instant  center  of  the  con- 
necting rod  relative  to  the  engine  frame  is  at  c.  If  VT,  the  tangential 
velocity  of  the  crank  pin,  is  known,  the  velocity  of  the  piston  and  cross- 


288 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


head  at  any  instant  may  be  found.  The  linear  velocity  at  any  point  in  a 
rigid  revolving  system  is  directly  proportional  to  its  distance  from  the 
center;  then: 

V_      Zi 

VT      12 
or: 


F-TV| 

62 


(7) 


^:>, 


.± 

v--«" 

FIG.  173. 

In  Fig.  173,  ceb  and  oea  are  similar  triangles;  then: 

li=  K 
Z2       R 
Substituting  in  (7)  gives: 

F=F,|- 
If  the  scale  of  velocity  is  such  that  VT  is  represented  by  R,  then  V  is  given 


FIG.  174. 

by  K,  which  may  be  measured  from  a  diagram  and  plotted  on  the  piston 
path  or  crank  circle,  the  latter  being  sometimes  rectified.     Fig.  174  shows 
the  velocity  plotted  on  the  crosshead  path  and  crank  circle. 
From  an  inspection  of  Fig.  173  it  may  be  seen  that: 

K  =  R  sin  e  ±  R  cos  0  tan  0. 

The  plus  sign  applies  to  the  half  of  the  crank  circle  toward  the  cylinder 
and  the  minus  sign  to  the  opposite  half.     Then : 

K 

5  =  sin  0  ±  cos  e  tan  <(>, 


THE  SLIDER  CRANK 


289 


Also : 


From  Fig.  173: 
From  which: 

Let 
then: 
and: 
Then: 

and: 
Then: 

LO/U   ti»    — 

COS  0 

L  sin  0  =  R  sin  0. 

#   . 

sin  0  =  ~Y  sin  0. 
L 

L 

~R  =  H' 

sin0 

n 

1              /fs\rt  fl\   2 

.                         /^                                 n       ,                        l-j                   /Sill    C7\ 

cos  <p  —  v  -i  —  sin   <p  —  A  /  11          i  * 
sin  0 

T  i  n  r/>  —                

n     1       fSm    *V 

N1       \    n    ) 

K      T,                cos  9          1  . 
—  =     1   +  —  sin  9 

fi               n   L       /sin  9\  2 

L           \         V   n   /  J 

,,,.,..    V  =  V^           :      : 
=  Fr  (sin  0  ±  cos  0  tan  </>) 

=    F.Fl                      C°Se               Ln» 

fU/1 


_  /sm 
\  n 


(8) 


(9) 


101.  Forces  Due  to  Pressure  in  Cylinder. — The  forces  acting  on  the 
slider-crank  mechanism  may  also  be  determined  by  the  method  of  instant 
centers  and  by  the  use  of  Fig.  175,  in  which  P  is  the  total  unbalanced 
pressure  on  the  piston  (see  notation). 

The  tangential  force  (also  known  as  turning  effort  and  crank  effort) 
due  to  P,  denoted  by  PT}  may  be  found  by  taking  moments  about  instant 
center  c;  or: 

Pli  =  PT  It. 

19 


290  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

From  which, 

p_  =P^. 
k 

By  similar  triangles  as  before : 


Then: 


li=K 
12    ''   R 

PT=P.- 

=  P(sin  0  ±  cos  6  tan 

cos  e 


=  PI  + 


L       /sm 
ttA/1  —  (- 
\         V   n 


sin  6\  2 


=  sin  6 


(10) 


FIG.  175. 

PT  is  a  positive  turning  effort  when  the  pressure  P  acts  in  the  same  di- 
rection as  the  movement  of  the  piston;  if  in  the  opposite  direction  it  is 
negative. 

The  force  due  to  P  transmitted  along  the  connecting  rod,  denoted  by 
PL,  is  found  by  taking  moments  about  shaft  center  o  and  substituting 
(10).  Then: 

PLlo  =  PrR  =  PK 
or: 


By  similar  triangles : 
K-          L 

lo 


n 


THE  SLIDER  CRANK  291 

Then: 


P    -PK 

PL~PTO 


(11) 


PL  is  the  maximum  force  due  to  P  producing  bending  moment  on  crank 
*  pin  and  shaft  at  any  time,  and  equals  P  at  dead  center. 

The  normal  force  on  guide  due  to  P  is  denoted  by  PN,  and  is  also  found 
by  taking  moments  about  shaft  center  o  and  substituting  (10).     Then: 

PNx  =  PTR 

or: 


.... 

By  similar  triangles  : 

1 
K  y  R  sin  6 


x       VL2-y2 
Then: 


Psinfl 


L       /sin 

n  V1  ~  (I 


The  direction  of  action  of  PN  is  shown  in  Fig.  204  and  Table  56. 

Force  P  is  transmitted  as  an  equal  and  parallel  force  to  the  crank 
pin,  which  in  turn  is  transmitted  to  the  main  bearing.  Two  couples 
then  hold  the  rod  in  equilibrium  and  may  be  equated  as  follows: 

PNL  cos  (f>  =  PL  sin  0 
or: 

PN  =  P  tan  0. 

It  is  obvious  from  Fig.  175  that  K/x  =  tan  0,  so  that  the  value  of  PN  is 
the  same  as  that  given  by  (12). 

The  radial  force  on  the  crank  due  to  P  is  denoted  by  PRj  and  is  found  by 
taking  moments  about  instant  center  a;  or: 

P«Z3  =  Prh 
and: 

PR=PTl^=  PT cot  (0  +  <*>). 


292  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

But: 


PT=  PL      =   PL  Sin  (0 


Then: 


PL  cos  (e  +  0) 
p  cos  (e  +  </>) 


=  p  cos  e  - 


sin20 


_  /sin  0\ 
\   n   / 


This  is  zero  when  6  +  <£  equals  90 — when  crank  and  connecting  rod  are 
at  right  angles. 

Efficiency  of  the  Slider  Crank. — From  (9)  and  (10)  it  is  clear  that: 

PTVT  =  PV  (14) 

It  is  shown  in  mechanics  that  PV  is  the  rate  of  work  done  at  any  instant, 
and  since  (14)  holds  good  for  every  position  occupied  by  the  mechanism, 
the  work  done  at  the  crank  pin  equals  that  done  in  the  engine  cylinder 
if  friction  is  neglected ;  or  the  efficiency  is : 


= 


(15) 


When  it  is  considered  that  in  high-grade  steam  engines  the  entire  loss  by 
friction  is  sometimes  less  than  5  per  cent.,  the  futility  of  trying  to  replace 
the  slider  crank  with  mechanisms  for  which  great  saving  is  claimed  is 
apparent. 

102.  Forces  Due  to  Gravity.  Horizontal  Engine. — The  connecting; 
rod  of  a  horizontal  engine  has  a  negative  effect  upon  the  turning  effort 
when  the  crank  is  in  the  half  of  the  crank  circle  toward  the  cylinder,  and. 
positive  when  on  the  opposite  side. 

TABLE  50 


Quadrant 

Force 

Formula 

1 

2 

3 

4 

i 

GT 

(16) 



+ 

•-h 



CT 

(18) 

— 

+ 

+ 

— 

THE  SLIDER  CRANK  293 

Let  GB  be  the  reaction  of  the  rod  of  weight  W  at  the  crank  pin.  The 
turning  effort  may  be  found  by  taking  moments  about  shaft  center  o; 
then : 

GTR  =  GBR  cos  6 
or: 

GT  =  GB  COS  6  =   Lf  W  COS  8  (16) 

Li 

The  force  GB  also  acts  as  an  equal  and  parallel  force  at  the  center  of 
the  main  bearing.  The  force  due  to  W  acting  on  the  guide  is: 

GA  =  W  -GB  (17) 

Let  C  be  the  weight  of  the  unbalanced  part  of  the  crank,  or  the  weight 
of  the  crank  pin;  or  if  a  counterbalance  is  used  it  may  be  the  unbalanced 


__  —  --M    // 

— - -L  "'"  \  /' 

--—  __^  Jjf        i 


FIG.  176. 


part  of  this  weight.  Let  rc  be  the  distance  from  the  center  of  the  shaft 
to  the  center  of  gravity  of  the  weight  C.  The  algebraic  sum  of  these 
weights  referred  to  the  crank  pin  center  may  be  expressed  by: 


8") 


Then  the  turning  effort  due  to  these  is: 

CT  =   2  (^  C  )  cos  0  (18) 

The  signs  for  GT  and  CT  for  the  different  quadrants  in  Fig.  176  are  given 
in  Table  50  under  the  corresponding  quadrant  numbers.  The  plus  sign 
indicates  a  positive  turning  effort — that  the  force  acts  in  the  direction  of 
motion;  while  the  minus  sign  is  for  a  negative  effect.  If  the  overbalance 
due  to  C  is  on  the  crank  side  the  signs  are  as  given  in  Table  50;  if  opposite 
the  crank  the  signs  for  CT  must  be  reversed. 

Aside  from  friction  the  weight  of  other  parts  does  not  affect  the  turn- 
ing effort. 


294 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Vertical  Engine. — The  weight  of  the  piston,  piston  rod  and  cross- 
head  obviously  affect  the  turning  effort,  but  as  it  acts  in  the  same  line 
with  the  pressure  it  may  be  accounted  for  in  drawing  the  indicator 
diagram. 

The  effect  of  the  connecting  rod  is  to  increase  the  turning  effort  on  the 
down  stroke  and  decrease  it  on  the  up  stroke;  its  weight  is  sometimes 

added  to  that  of  the  strictly  reciprocating 
parts  but  it  is  more  correctly  treated  by 
itself  and  this  will  be  done,  leaving  the 
approximate  method  to  the  judgment  of 
the  designer. 

The  rod  exerts  a  downward  force  G  and 
a  lateral  force  GN  as  shown  in  Fig.  177.  It 
must  be  assumed  that  the  entire  weight  is 
supported  by  the  crank  pin;  then: 

G  =  W. 

The  weight  of  the  rod  acting  at  the 
center  of  gravity  and  resisted  by  an  equal 
force  at  the  crank  pin  forms  a  couple  equal 
to: 

G(L  —  LG)  sin  0. 

This  is  balanced  by  the  couple: 

GNL  cos  0 
or: 

GNL  cos  <t>  =  W(L  —  L0)  sin  </>. 

From  which: 

T,~    sin  th 

GN  ~- 


(19) 


wn  - 


sin  0 


FIG.  177. 


/sn  0\ 

~  hr) 


Taking  moments  about  crank  center  o 

for  forces  G}  GN  and  C,  the  corresponding  turning  efforts  are  found. 
The  equations  are: 

GXTR  =  GR  sin  0 
GNTR  =  GNR  cos  0 


CTR  = 


THE  SLIDER  CRANK 


295 


Then  by  cancellation  and  substitution  the  turning  efforts  are: 

GXT  =  W  sin  6 

LG\  sin  26 


GNT  =GNCOSe=w(l-£G} 


^C  Bin* 


The  total  turning  effort  due  to  the  gravity  of  the  rod  is: 

GT  =  GXT  +  GNT 
TABLE  51 


(20) 
(21) 

(22) 
(23) 


Quadrant 

Force 

Formula 

1 

2 

3 

4 

GXT 

(20) 

+ 

+ 





GNT 

(21) 

-f 

— 

+ 

— 

CT 

(22) 

+ 

+ 

— 

— 

The  signs  for  the  quantities  GXT,  GNT  and  CT  for  crank  positions  in  the 
different  quadrants  of  Fig.  177  are  given  in  Table  51,  the  plus  sign  for 
forces  acting  in  the  direction  of  turning  as  . 

before;   the   sign  for   CT  assumes  the  over-        ,..--" 
balance  on  the  pin  side  of  the  crank.     The    t    A 
forces  G  and  GN  also  act  at  the  center  of  the 
main  bearing. 

If  the  engine  center  line  is  inclined  from 
the  vertical  5  degrees  as  shown  in  Fig.  178, 
modification  of  the  reactions  must  be  made  as 
follows  : 


GB  =  W  -  W(l  -      )  sin  5 


(24) 
(25) 


GN  =G(l  -  -^)  tan  <f>  cos  5 
The  effect  on  the  turning  effort  is: 

GT  =  GB  sin  (0  -  d) 

GNT  =  GN  COS  (0  —  6) 
CT  =  s 


FIG.  178. 


(26) 

(27) 
(28) 


296 


DESIGN  AND  CONSTRUCTION  OF  PI  EAT  ENGINES 


The  signs  for  the  different  portions  of  the  crank  pin  path  depend  upon 
the  angle  5  and  may  be  determined  by  inspection  for  a  particular  case. 
Formulas  (24)  to  (28)  reduce  to  those  already  given  for  a  horizontal 
or  vertical  engine. 

103.  Forces  Due  to  Acceleration. — Practically  all  the  inertia  forces 
of  the  slider-crank  mechanism  may  be  accounted  for  by  considering  the 
acceleration  of  any  point  in  the  connecting  rod  in  the  direction  x,  parallel 
to  the  crosshead  path,  and  the  direction  y  normal  to  the  crosshead  path. 
In  the  formulas,  which  will  be  given  without  derivation,  co  is  the  angular 
velocity  in  radians  per  second,  and  may  be  expressed  in  terms  of  r.p.m. 
thus: 

2irN       irN 

"GOT    =30  (2°) 

The  acceleration  in  ft.  per  sec.  per  sec.  in  the  direction  x,  Fig.  179  is: 

2  0 


ax 


on   . 
4-  cos  26  + 

- 


and  in  the  direction  y: 


a'=I 


sin  6 


(30) 


(31) 


FIG  179. 


As  force  is  the  product  of  mass  and  acceleration,  the  general  formula  is : 

(32) 

in  which  a  may  be  either  ax  or 
aY,  and  if  W  is  the  total  weight, 
a  must  be  the  acceleration  of 
the  mass  center.  Formulas  (30) 
and  (31)  give  acceleration  in 
any  point  in  the  rod;  and  for 
any  mass  assumed  as  concen- 
trated at  that  point,  (32)  gives  the  force.  An  approximation  to  (30) 
is  given  by  (48),  the  latter  being  much  more  simple,  and  for  most  practical 
applications  accurate  enough. 

Hereafter  FY  will  denote  the  inertia  of  the  rod  normal  to  the  line  of 

stroke,  FYA  its  effect  at  the  crosshead  and  FYB  at  the  crank.    Fx  will  denote 

the  inertia  of  the  rod  along  the  line  of  stroke,  referred  to  its  mass  center. 

When  I  is  zero,  ax  is  the  acceleration  of  the  piston  and  other  purely 

reciprocating  parts.     The  inertia  of  these  parts  will  be  denoted  by  FP. 

The  force  exerted  by  the  inertia  of  a  mass  upon  the  engine  parts  in  a 


THE  SLIDER  CRANK  297 

horizontal  direction  is  toward  the  cylinder  when  the  crank  is  on  this  side 
of  the  crank  circle,  and  in  the  opposite  direction  when  the  crank  is  on 
the  side  away  from  the  cylinder;  an  exception  is  when  the  crank  is  in  the 
vicinity  of  90  degrees  from  the  engine  center  line,  and  this  may  be  seen 
by  plotting  (30)  as  in  Fig.  180,  in  which  aH  and  ac  are  accelerations  at 
head  and  crank  ends  of  the  stroke  respectively. 

The  signs  in  Formula  (30)  are 
used  simply  to  obtain  numerical 
results,  and  for  this  purpose  may 
be  taken  as  plus  when  angles  6  and 
26  are  in  the  half  of  the  crank 
circle  toward  the  cylinder.  This 

has  a  negative  effect  upon  the  turning  effort  on  the  stroke  from  head 
to  crank  end  when  the  numerical  result  has  a  positive  sign,  and  a  posi- 
tive result  when  the  sign  is  minus.  On  the  return  stroke,  however,  the 
effect  is  just  the  reverse.  This  may  be  shown  by  Fig.  181,  which  gives 
the  acceleration  diagrams  for  the  two  strokes,  the  positive  sign  de- 
noting that  the  inertia  of  the  moving  part  would  assist  the  turning 
effort  and  the  negative  sign  that  it  would  have  a  retarding  effect. 

The  Reciprocating  Parts. — If  I  equals  L  in  (30),  ax  is  the  acceleration 
of  the  crosshead  and  piston.     At  the  dead  centers  (30)  becomes :   » 

ax  =  R<. 

The  minus  sign  gives  the  numerical  value  for  the  head  end  and  the  plus 
sign  for  the  crank  end.     Then  the  inertia  of  the  reciprocating  parts  for 
head  and  crank  ends  may  be  computed,  and  other  points  found  graphi- 
cally by  Klein's  method  shown  in  Fig.  182. 
From  (32) : 


(33) 

A\JO\J       \  'I If  / 

and: 


2930 


298 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Fig.  182  is  an  exact  representation  of  (30)  and  proof  is  given  in  Dalby's 
Balancing  of  Engines  previously  referred  to.  The  lower  part  of  the  con- 
struction is  added  for  the  convenience  of  plotting  to  any  scale.  After 
deciding  upon  a  force  scale,  FH  and  Fc  may  be  laid  off  on  the  diagrams 
as  shown ;  the  lines  ab  and  cd  are  located  by  making  the  construction  at 
the  dead  center — only  one  of  these  being  necessary. 

The  Connecting  Rod. — If  in  (30),  I  equals  LG,  the  distance  to  the  mass 
center,  the  effect  of  acceleration  of  the  rod  parallel  to  the  line  of  stroke 
may  be  found.  If  we  assume  n0  an  imaginary  value  of  n,  and  substitute 
in  (30): 

n 

n0  -- 


FIG.  182. 

then  using  this  value  in  (33)  and  (34),  values  of  Fx  for  the  rod  may  be 
found  graphically. 

The  inertia  of  the  reciprocating  parts  may  be  combined  with  the 
indicator  diagram  of  total  pressure  before  determining  turning  effort, 
and  often  the  connecting  rod  is  added  to  the  reciprocating  parts  to 
simplify  calculation,  probably  with  sufficient  accuracy  in  most  cases; 
but  to  be  more  exact  the  rod  requires  a  separate  treatment,  and  FX)  the 
inertia  of  the  rod  assumed  concentrated  at  the  mass  center  may  be  treated 
as  W  in  connection  with  the  gravity  effect  of  the  rod  for  vertical  engines. 
Then  for  the  component  parallel  to  line  of  stroke,  comparing  with  (20): 

FXT  =  Fx  sin  0  (35) 

For  the  component  normal  to  line  of  stroke,   comparing  with   (21): 

L0\  sin  20 

(36) 


THE  SLIDER  CRANK 


299 


Fig.  183  shows  the  effect  of  rod  inertia;  it  may  also  be  used  as  a 
diagram  for  the  effect  of  inertia  of  the  reciprocating  parts,  taking  FP  only, 
if  it  is  desired  to  treat  it  separate  from  the  indicator  diagram.  The  total 
effect  of  Fx  on  the  turning  effort  is: 

(37) 


XT 


XNT 


TABLE  52 


Sector  of  circle 

• 

T?                    1 

1 

2 

3 

4 

5 

6 

•Fxr 

(35) 



+ 

+ 





+ 

FXNT 

(36) 

— 

+ 

— 

+ 

— 

+ 

The  signs  of  FXT  and  FXNT  are  given  in  Table  52  for  different  sectors 
of  the  circle  as  shown  in  Fig.  183.     A  change  occurs  when  the  inertia 


FIG.  183. 

curve  crosses  the  line  in  the  case  of  FXNT,  and  again  when  the  crank  is  at 
right  angles  to  the  center  line  of  the  engine. 

The  effect  of  angular  acceleration  is  best  treated  by  finding  the  moment 
of  inertia  as  described  in  Par.  99.  To  derive  the  formulas,  assume  the  rod 
divided  into  a  number  of  parts  of  weight  w,  each  a  distance  I  ft.  from  the 
center  of  the  crosshead  pin.  The  force  due  to  acceleration  of  the  rod 
normal  to  the  line  of  stroke  acts  away  from  the  center  line  of  engine,  and 


300 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


its  effect  on  turning  effort  is  shown  in  Fig.  184.     The  force  exerted  by 
each  section  of  the  rod  is : 

FY  =  —  aY  =  —  'T-Rw*sm  9  (38) 

g        g  L 

The  reaction  of  this  force  on  the  crank  pin  in  a  direction  normal  to  the 
line  of  stroke  is : 


The  total  reaction  is: 


FYB  =  2FYN  =  — r  •  2wZ2  •  sin  9 


CO 


T 

gnL 


•  /  sin  9 


FIG.  184. 

The  effect  upon  the  turning  effort  is: 
FYT  =  FYB  COS  6 

..2  '1 

•  7  sin  29 


2gnL 

N* 


5865nL 


.  7  sin  29 


(39) 


(40) 


The  signs  for  FYT  for  the  different  quadrants  are  given  in  Fig.  184. 
The  total  turning  effect  of  the  rod  inertia  is : 


FT  —  FXT  ~\-  FXNT  +  FYT 


(41) 


The  reaction  on  the  guide  due  to  FY  is  found  by  taking  moments  about 
the  center  of  percussion,  the  location  of  which  is  given  by  (6).  If  FYA 
is  this  reaction: 


THE  SLIDER  CRANK 


301 


From  which: 


(42) 


The  reaction  on  the  guide  due  to  Fx  is  found  as  for  W  in  (19);  or: 

sin  6 


F 


XN 


/sin  0\ 

"  br) 


The  total  guide  pressure  due  to  acceleration  of  the  rod  is: 

FA  —  FYA  ~\~  FXN 


(43) 


(44) 


The  forces  FXN,  Fx  and  FYB  acting  at  the  crank  pin,  also  act  as  equal 
and  parallel  forces  at  the  center  of  the  shaft. 

104.  Combined  Indicator  and  Inertia  Diagrams. — As  the  pressure 
on  the  piston  and  inertia  of  the  reciprocating  parts  are  both  applied  to 
the  crosshead  pin,  they  may  be  combined  and  their  turning  effort  con- 
sidered together.  It  is  first  necessary  to  construct  a  stroke  diagram 
giving  the  unbalanced  steam  or  gas  pressure;  that  is,  the  difference 
between  the  pressure  on  the  two  sides  of  the  piston.  This  is  found  by 
plotting  the  distance  between  the  forward  pressure  of  one  diagram  and  the 
back  pressure  of  a  diagram  for  the  other  side  of  the  piston  taken  at  the 
same  time. 

Steam  engine  diagrams  will  be 
first  considered.  A  pair  of  indi- 
cator diagrams  is  shown  in  Fig.  185, 
and  it  is  best  that  they  should  be 
diagrams  of  total  pressure  as  this 
allows  for  the  piston  rod.  The 
shaded  area  represents  the  stroke- 
diagram  for  the  head  end,  giving  the 
pressures  acting  on  the  piston  during 
the  stroke  from  head  to  crank  end  of  FIG.  185. 

cylinder.     The  pressure  is  zero  at  a 

and  becomes  negative  after  that.  For  convenience  this  may  be  plotted 
from  a  straight  line,  and  taking  pressures  above  the  plotting  line  as 
driving  the  crank  in  the  direction  of  motion  (with  the  plus  sign),  and 
below  the  line  as  tending  to  retard  the  motion  of  the  crank  (with  the 
minus  sign),  Fig.  186  is  plotted,  with  inertia  diagram  shown  below,  and 
finally  the  combined  diagrams. 

A  convenient  method  of  plotting  the  combined  diagrams  is  to  invert 
the  inertia  diagram  on  the  stroke  diagram  as  shown  in  dotted  lines,  thus 


302 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


performing  graphical  addition  and  subtraction;  then  the  measurements 
may  be  transferred  to  a  diagram  plotted  from  a  straight  line,  or  may  be 
taken  directly  to  plot  a  crank  effort  diagram.  This  applies  especially 
to  steam  engine  diagrams;  for  the  internal-combustion  engine  confusion 
may  be  avoided  by  following  the  method  shown  by  the  full  lines,  adding 
the  pressures  algebraically. 

If  the  engine  is  vertical  the  weight  of  the  reciprocating  parts  should 
be  added  to  the  down-stroke  diagram  and  subtracted  from  the  up-stroke 


FIG.  186. 


diagram  as  shown  in  dotted  lines;  Fig.  186  shows  the  addition  made  to 
the  head-end  diagram,  which  would  be  the  case  if  the  cylinder  were 
above  the  crank — the  most  common  arrangement.  If  the  center  line 
of  the  engine  leans  8  degrees  from  the  vertical  as  in  Fig.  178: 


Pw  *=  W  cos  5 


(45) 


This  quantity  is  added  to  the  head  end  if  the  cylinder  is  higher  than  the 
shaft,  and  subtracted  from  the  crank  end. 


THE  SLIDER  CRANK 


303 


Internal-combustion  Engine  Diagrams. — The  conventional  stroke  dia- 
gram for  a  4-cycle  constant- volume  engine  is  shown  in  Fig.  187  with 
inertia  diagram  and  combined  diagram  below.  The  diagram  is  for  an 
automobile  engine  running  1500  r.p.m.  The  connecting  rod  is  not 
included  in  the  inertia  diagram,  so  that  the  combined  diagram  shows 
only  the  forces  applied  at  the  wrist  pin.  The  inertia  of  the  rod  must  be 
included  in  the  forces  acting  at  the  crank,  but  this  is  best  given  a  separate 
treatment.  Data  for  the  diagrams  are  given  in  Table  55.  Accurate 
results  cannot  be  obtained  by  including  the  rod  with  the  reciprocating 
parts,  especially  if  the  center  of  gravity  is  near  the  crank  pin,  which  is 
true  of  this  type.  In  modern  well-designed  engines,  suction  and  exhaust 
pressures  differ  but  little  and  both  are  assumed  as  atmospheric  in  this 
chapter. 


FIG.  187. 

If  the  engine  is  not  horizontal,  Pw  (shown  dotted)  must  be  taken  into 
account  as  in  Fig.  186  and  Formula  (45). 

In  finding  values  of  P  for  different  crank  positions,  the  spacing  around 
the  crank  circle  should  be  equal;  the  corresponding  piston  positions  are 
therefore  not  equally  spaced  and  the  spaces  are  greater  at  the  head  end 
than  at  the  crank  end  of  the  stroke.  These  points  in  the  piston  path  may 
be  found  as  in  Fig.  188,  then  marked  on  a  piece  of  tracing  cloth  and 
transferred  to  diagrams  such  as  Figs.  186  and  187  with  a  pricker  point, 
placing  the  tracing  cloth  end-for-end  for  the  return  strokes  in  order  that 
letters  H  and  C  may  fall  on  the  same  letters  on  the  diagram. 

Figs.  186  and  187  show  that  pressure  is  constantly  exerted  against  the 
bearings,  absorbing  work  by  friction.  By  plotting  the  forces  of  such 


304 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


diagrams  upon  a  rectified  crank  circle  the  mean  pressure  upon  crank  pin 
and  shaft  may  be  obtained  for  the  cycle.  An  approximation  may  be 
made  by  including  the  connecting  rod  with  the  reciprocating  parts. 
Then  the  mean  pressure  from  such  diagrams  is  a  measure  of  the  work 
absorbed  by  friction;  this  is  not  the  mean  effective  pressure  or  the  alge- 


FIG.  188. 

braic  sum  of  the  pressures,  but  the  mean  of  the  actual  pressure  against 
the  bearings  during  the  cycle,  regardless  of  signs.  For  the  main  bearing 
a  resultant  should  be  taken,  including  weight  of  wheel,  shaft,  etc.,  pull 
of  belt  or  reaction  due  to  gears,  or  any  other  forces  acting  on  the  bearing. 
105.  Reversal  of  Thrust  and  Effect  of  Compression. — It  has  been 
generally  accepted  that  compression  is  necessary  to  smoothness  of 
operation  of  a  steam  engine,  but  in  few  instances  is  it  probable  that  the 
necessary  compression  has  been  fixed  by  calculation,  or  if  so,  the  inertia 
of  the  reciprocating  parts  has  probably  been  neglected.  An  examination 

of  this  is  best  made  by  inverting 
the  crank  side  of  the  upper  dia- 
gram of  Fig.  186.  The  pressures 
above  the  plotting  line  are  those 
acting  toward  the  crank,  while  those 
below  the  line  act  toward  the  head 
end.  This  is  shown  in  Fig.  189. 
Neglecting  inertia,  reversal  of  thrust 


occurs  when  the  pressure  line  crosses 
the  plotting  line,  which,  for  the  dia- 
grams shown  gives  a  gradual  re- 
versal, occurring  before  the  end  of 
the  stroke.  When  inertia  is  con- 
sidered it  occurs  where  the  pressure  line  crosses  the  inertia  curve;  this 
curve  may  be  considered  the  virtual  plotting  line  for  the  remaining  dia- 
grams of  this  paragraph. 

The  effect  of  inertia  is  to  cause  reversal  to  occur  later  in  the  stroke, 
but  in  Fig.  189  it  is  still  gradual*;  if  the  diagrams  were  plotted  on  a  rectj- 


FIG.  189. 


THE  SLIDER  CRANK  305 

fied  crank  circle — a  more  correct  way  of  determining  the  relative  time  of 
reversal — it  would  appear  still  more  gradual. 

With  higher  speed  assume  the  inertia  curve  to  be  the  dotted  line  A. 
In  any  case  reversal  occurs  where  the  inertia  curve  crosses  the  pressure 
curve,  therefore  for  curve  A  the  reversal  is  at  initial  pressure  for  the  head 
end  and  a  little  less  at  the  crank  end.  The  reversal  at  head  end  is  gradual, 
and  at  crank  end  the  reversal  force  is  the  pressure  represented  by  ran, 
and  while  it  appears  to  occur  abruptly,  it  is  probable  that  the  crank 
travels  over  an  appreciable  angle  while  the  pressure  is  being  applied  and 
there  would  be  little  or  no  tendency  to  "  pound, "  even  with  some  play 
in  the  pin  bearings.  However,  if  the  inertia  is  less  and  the  curve  does 
not  cut  the  pressure  curve  (due  to  compression)  an  appreciable  distance 
before  the  dead  center,  a  pound  may  be  heard  if  the  pin  bearings  are  not 
properly  adjusted. 

With  inertia  curve  B,  reversal  at  head  end  occurs  at  o,  after  the  stroke 
has  begun,  but  is  gradual  and  would  probably  cause  no  pound;  at  crank 
end  it  occurs  at  dead  center  but  is  gradual. 

If  there  were  no  compression  the  pressure  curve  would  be  as  shown 
clotted;  then  for  the  inertia  curve  shown  in  full  lines  the  force  due  to- 
re versal  would  change  at  dead  center  from  ql  to  In;  for  curve  A  it  is 
from  qm  to  mn,  and  for  curve  B  from  qn  to  zero,  the  reversal  force  being 
gradually  applied.  It  is  obvious  that  if  reversal  does  not  occur  before 
the  end  of  the  stroke,  the  force  ql,  qm  or  qn  has  no  influence  upon  smooth- 
ness of  running,  but  that  this  depends  upon  the  magnitude  of  the  force 
starting  the  piston  on  its  return  stroke,  such  as  In  or  ran;  if  this  is  large 
a  " knock"  or  " pound"  may  be  the  result,  and  it  is  clearly  an  advantage 
when  inertia  is  small,  to  have  the  pressure  curve  cross  the  inertia  curve 
before  the  end  of  the  stroke.  A  serious  obstacle  to  this  is  that  without  a 
high  compression,  the  terminal  pressure  at  large  loads  is  higher  than  the 
compression  pressure  and  the  pressure  line  does  not  cross  the  plotting  line; 
this  is  especially  true  with  single-eccentric,  non-releasing  gears,  which 
reduce  the  compression  as  the  terminal  pressure  is  increased.  The 
uniflow  engine  has  some  advantage  in  this  respect,  but  with  heavy  loads 
reversal  may  occur  but  little,  if  any  before  the  end  of  the  stroke. 

A  study  of  the  diagrams  makes  clear  the  fact  that  increasing  the 
weight  of  the  reciprocating  parts  has  sometimes  aided  in  smoothness 
of  operation;  it  may  also  account  for  the  non-analytical  discussions 
in  technical  papers  from  time  to  time  by  operating  engineers,  as  to  the 
necessity  of  compression,  with  fervent  adherents  to  both  sides  of  the 
question,  which  may  only  be  satisfactorily  explained  by  the  use  of  re- 
versal diagrams. 

20 


306  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

As  the  piston  speed  S  =  4RN,  Formulas  (33)  and  (34)  may  be  written: 


This  shows  the  influence  of  weight,  piston  speed  and  rotative  speed  upon 
inertia.  To  obtain  the  reversal  pressure  at  the  crosshead  pin,  the 
reciprocating  parts  proper  must  be  considered  alone;  for  the  effect  at  the 
crank  pin,  it  will  be  approximately  correct  to  add  the  weight  of  the  con- 
necting rod  to  W,  but  more  accurate  to  determine  the  inertia  of  the  rod 
separately  as  explained  in  Par.  103. 

When  an  engine  is  running  idle,  as  a  locomotive  in  "coasting," 
reversal  curves  are  due  to  inertia  only,  and  are  shown  in  Fig.  190.  Rever- 
sal occurs  near  mid-stroke  and  is  very  gradual.  Indicator  diagrams  for  a 
single-acting  engine  are  also  shown  in  Fig.  190  with  dotted  lines.  As  the 
pressure  line  does  not  cross  the  inertia  curve  at  the  crank  end  of  the  stroke, 
there  is  no  reversal  of  thrust  at  this  place;  on  the  return  stroke  reversal 


J  » » 

FIG.  190. 

occurs  when  the  compression  curve  crosses  the  inertia  curve,  and  unless 
inertia  is  nearly  equal  to  or  greater  than  initial  pressure,  there  is  advan- 
tage in  having  the  compression  pressure  greater  than  the  inertia.  It 
should  be  noted  that  the  reversal  which  would  have  occurred  at  the 
crank  end  of  the  stroke  in  a  double-acting  engine,  is  deferred  until  the 
inertia  curve  crosses  the  plotting  line  on  the  return  stroke  a  short  time 
before  the  head-end  reversal. 

It  may  be  seen  from  Fig.  189  that  if  compression  pressure  is  high 
relative  to  inertia,  the  effect  of  inertia  is  more  largely  applied  at  the 
cylinder  end  of  the  frame;  whereas  with  high  inertia  pressure  and  small 
compression  the  retarding  effect  comes  on  the  bearing. 

In  Fig.  189,  if  the  cut-off  is  long,  as  shown  dotted,  the  turning  effort 
will  be  as  uniform  as  possible  if  the  inertia  effect  is  negligible;  the  shorter 
the  cut-off  the  less  uniform  it  becomes.  If  the  inertia  effect  is  increased 
for  the  short  cut-off  the  effective  pressure  becomes  more  uniform  through- 
out the  stroke.  A  locomotive  may  thus  be  started  with  a  maximum 
cut-off,  and  by  bringing  back  the  reverse  lever  gradually  as  the  engine 
gains  speed,  the  maximum  uniformity  may  be  obtained  at  all  times. 
Failure  to  do  this  results  in  discomfort  to  passengers  and  increased  wear 
and  tear  on  the  mechanism. 


THE  SLIDER  CRANK 


307 


A  steam  automobile  will  take  a  hill  more  smoothly  when  running 
slow,  if  the  cut-off  be  lengthened  and  the  pressure  throttled. 

A  single-acting,  constant- volume  internal-combustion  engine  diagram 
is  shown  in  Fig.  191.  For  the  full  lines  there  is  no  reversal  except  where 
the  inertia  curve  crosses  the  plotting  line  during  the  exhaust  and  suction 
strokes.  Should  inertia  at  the  head  end  of  the  stroke  be  high  enough  to 
cross  the  compression  line  as  shown  dotted,  reversal  occurs  gradually 


EXPANSION 


EXHAUST 


SUCTION 
FIG.  191. 


COMPRESSION 


where  the  lines  cross  at  n,  and  again  at  the  end  of  the  stroke.  The  mag- 
nitude of  the  reversing  force  depends  upon  the  difference  between  inertia 
and  the  maximum  gas  pressure;  if  inertia  is  higher  as  in  curve  A,  Fig. 
191,  reversal  occurs  after  the  expansion  stroke  has  begun,  and  is  gradual. 
A  reversal  diagram  of  a  4-cycle,  double-acting  internal-combustion 
engine  is  shown  in  Fig.  192,  and  with  a  different  order  of  firing  in  Fig. 
193.  In  Fig.  192,  for  the  values  assumed,  reversal  occurs  at  the  begin- 
ning of  the  head-end  and  crank-end  expansion  strokes,  and  again,  gradu- 


H.EXP. 
C.  COM  P. 


H.SUCT. 
C.  EXH. 


H.  COM  P. 
C.SUCT.. 


FIG.  192. 


ally,  during  the  head-end  suction  stroke.  In  Fi5.  193,  abrupt  reversal 
occurs  only  at  the  beginning  of  the  head-end  expansion  stroke,  and  gradual 
reversal  during  the  head-end  exhaust  stroke.  Increasing  the  inertia  to 
the  dotted  lines  improves  conditions  for  Fig.  192  by  making  the  reversal 
gradual  near  the  end  of  head-end  compression,  and  reducing  the  reversal 
pressure  at  the  beginning  of  crank-end  expansion. 


308 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


s 

1 

10 

c 

CO 

2 

CO 

0 

^ 

o 

T-H 

r-  1 

r— 

T—  i 

0 

O 

0 

0 

O 

O 

o 

c 

d 

1 

1 

1 

§1 

C^ 

CO 

' 

10 

s 

05" 

s. 

t> 

00 

8 

§ 

8 

8 

T-H 

<N 

<M 
(N 

0 

»o 

a- 
i^ 

00 

•—  ' 

0 

0 

0 

0 

,_, 

,_( 

rH 

0 

0 

0 

0 

0 

0 

0 

c 

0 

1 

1 

1 

CO 

T-H 

^ 

i 

o 

o 

i 

00 

00 

g 

8 

8 

8 

g 

T-H 

^ 

(M 

T-H 

0 

T—  1 

0 

s 

& 

00 

!H 

0 

0 

o 

0 

0 

T— 

i-H 

0 

0 

0 

0 

0 

o 

0 

C 

0 

1 

1 

1 

O5 

Jv^ 

O5 

T-H 

T-H 

T-H 

05 

8 

t^ 

T—  1 

o 

05 

01 

05 

3 

CO 

CO 

T-H 

«-J 

1—  ( 

1> 

0 

0 

0 

0 

0 

o 

O 

O 

0 

o 

0 

o 

C 

3 

o 

1 

1 

1 

00 

o 

00 

g 

So 

g 

s 

8 

§ 

CO 

oo 

g 

<N 

CO 
CO 

s 

s 

i 

1 

£8 

T—  1 

d 

d 

d 

d 

d 

0 

0 

d 

o 

o 

0 

0 

o 

0 

c 

1 

O 

1 

1 

1 

O 

O5 

a 

i 

So 

oT 

i 

i 

i 

§ 

§ 

O5 

§ 

CO 

T-H 

00 

Cf 

*3 

5 
1 

0 

0 

o 

0 

0 

0 

c 

0 

0 

0 

0 

0 

0 

0 

d 

c 

5 

0 

1 

1 

oo 

^44 

O5 

CO 

i 

T-H 

0 

0 

T-H 

O5 

o 

c 

d 

T-H 

T-H 

T-H 

d 

d 

CO 

d 

a 

d 

g 

c 

» 
) 

T-H 

d 

1 

1 

10 

10 

S 

a 

g 

00 

£ 

oc 

a- 

§1 

§ 

§ 

T*H 

o 

1 

g 

s 

s 

2 

\ 

T—  1 

o 

0 

0 

o 

0 

c 

0 

T-H 

T-H 

rH 

0 

0 

0 

0 

G 

> 

0 

* 

i 

b 

g 

00 

0 

8 

g 

O5 

8 

§ 

s 

(N 

§ 

2 

oo 

CO 

c 

1 

) 
t 

<N 

o 

d 

d 

0 

d 

c 

d 

d 

d 

d 

d 

0 

o 

0 

c 

) 

d 

CO 

i? 

T-H 

£ 

^ 

0 

s 

g 

i 

S 

s§ 

T-H 

00 

S 

g 

CO 

T-H 
T-H 

T-H 

r- 

t- 

4 

T-H 

o 

0 

0 

c 

o 

o 

o 

0 

0 

0 

o 

o 

c 

> 

0 

CN 

s 

s 

So 

So 

g 

s? 

i 

' 

i 

T-H 

CO 

g 

Jo 

S 

0 

i 

g 

c 

5 

T—  1 

0 

d 

d 

0 

o 

0 

i 

^ 

d 

0 

0 

0 

o 

o 

T-H 

T- 

4 

T—  1 

10 

a 

S 

g 

00 

8 

i 

i 

8 

CO 

T-H 

CO 

s 

s 

1 

CO 

T-H 

tf 

5 

4 

CO 

0 

0 

0 

0 

r« 

- 

i 

•H 

o 

o 

0 

0 

0 

o 

T-H 

t* 

4 

~ 

« 

4 

1C 

> 

CO 

^ 

10 

CO 

* 

»o 

CO 

« 

W 

3 

CO 

r 

M 

~l 

*s 

1 

°t» 

^  

\ 

^" 

i- 

*SK 

^ 

-~* 

\            ^ 

•—  -  .^, 

1. 

^ 

s 

-M 

Si 

5 

I 

* 

'53 

v^— 

^ 

S 

^L 

•—  •- 

X 

+1 

'a 

1 

H 

H 

™ 

«> 

« 

* 

• 

L 

J 

^» 
^-   — 

1 

^ 

i 

i 
j 

v^l 

- 

II 

* 

! 

9 

v° 

Q> 

_g 

3 

§g 

a 

w 
~  — 

^ 

S 

H 

i 

THE  SLIDER  CRANK 


309 


0 

to 

? 

^ 

C^j 

CO 

o 

l-H 

0 

l-H 

l-H 

!"H 

rH 

o 

O 

O 

o 

o 

o 

^ 

*~I 

* 

TH 

^- 

i-H 

CO 

tt 

a 

& 

o 

oo 

8 

i 

8 

s 

rH 

CO 

o 

i 

1 

co 

rH 

1C 

> 

i 

CO 

CO 

o 

o 

o 

o 

T-H 

T-H 

l-H 

0 

0 

o 

o 

0 

o 

rH 

r- 

•H 

__ 

0 

0 

on 

oo 

o 

O5 

O5 

8 

8 

CO 

O5 

to 

Jo 

8 

i-H 

o 

i 

g 

a 

0 

<N 

& 

b 

b 

b 

b 

b 

rH 

l-H 

O 

b 

b 

b 

b 

b 

rH 

T— 

- 

rH 

to 

£ 

£ 

s 

O: 
C: 

O5 

co 

rH 

00 

2 

i 

5 

O5 
i-H 

£ 

^ 
l> 

rH 

<M 

CO 

b 

b 

b 

C 

O 

b 

o 

b 

b 

b 

b 

b 

C 

i 

b 

8 

0 

So 

S 

SB 

s 

S 

C: 
o-. 

s 

a 

s 

3 

1 

$ 

S 

% 

c 

T^ 

> 

^ 

(N 

CO 

0 

0 

0 

0 

0 

C 

0 

o 

o 

0 

o 

0 

b 

b 

c 

) 

b 

05 

s 

0 

8 

§ 

S5 

& 

8 

§ 

§ 

§ 

i—  i 

0 

O5 

§ 

CO 

S 

c; 

i 

(N 

<N 

0 

0 

0 

0 

0 

C 

0 

l-H 

rH 

i-H 

0 

o 

0 

0 

c 

> 

0 

GQ 

^ 

OS 

00 

o 

l-H 

0 

o 

i-H 

& 

£ 

i 

TH 

rH 

,_, 

8 

a 

CO 

^ 

g 

> 

i 

rH 

C-l 

0 

C 

o 

b 

b 

b 

0 

c 

> 

0 

1 

1 

O5 

cp 

^ 

^ 

to 

to 

0 

8 

s 

So 

O5 

cr 

s 

i 

§8 

S 

05 
rH 

CO 

TH 

00 

9 

i 

S 

<N 

0 

0 

0 

o 

o 

c 

0 

o 

0 

0 

0 

0 

0 

0 

C 

> 

0 

1 

1 

CO 

0 

00 

s 

S5 

§ 

s 

? 

'     * 
1 

§1 

£ 

B 

t^ 

(N 

CO 

CO 

rH 

rH 

"s 

cl 

5 

s 

W 

0 

0 

0 

0 

0 

c 

0 

0 

o 

0 

0 

0 

0 

0 

C 

> 

0 

1 

1 

OS 

CO 

05 

to 

rH 

1—  1 

00 

c- 

O5 

00 

S 

C<l 

•^ 

rH 

T—  1 

T- 

4 

rH 

10 

0^ 

l-H 

o 

OS 

^ 

t 

CO 

I~H 

w 

0 

o 

0 

c 

l' 

0 

0 

0 

o 

o 

o 

O 

O 

C 

> 

O 

| 

1 

"* 

0 

g 

00 

oo 

s 

s 

i 

i 

8 

CO 

3 

58 

s 

i-H 

0 

i-H 

i 

g 

I~ 

b 

B 

c^ 

0 

0 

0 

o 

o 

T™ 

i-H 

0 

0 

0 

0 

0 

0 

O 

C 

> 

o 

1 

1 

to 

(N 

CO 

co 

to 

a 

0 

§ 

00 

8 

8 

8 

o 

rH 

(N 

c3 

8 

0 

to 

g 

> 

(N 

00 

T-H 

o 

0 

0 

0 

rH 

T— 

i-H 

0 

0 

0 

0 

0 

0 

0 

C 

> 

0 

1 

1 

r 

<£> 

C1 

<Z~ 

-1 
—  ^ 

.S 

s 

-— 

—  w 

8 

x  — 
,    1 

T" 

*# 

M 

s>*— 

•*s 

N 

<3s 

* 

-?> 

G 

£ 

S 

** 

a 

5 

g 

o 

"m 

-~  " 

s 

+1 

•*f. 

^  — 

~s 

H 

q 

-1 

> 

* 

(N 

<M 

L 

_J 

r— 

1 

_i 

\ 

£ 

« 

1 

a 

o5 

a 

— 

r 

*W 

4 

H 

-\ 

310 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


In  Fig.  193,  while  the  reversal  pressure  at  the  beginning  of  head-end 
expansion  is  reduced  by  increased  inertia,  an  abrupt  reversal  is  caused  at 
the  beginning  of  crank-end  expansion.  This  would  indicate  that  the 
smoother  order  of  firing  may  only  be  predetermined  by  the  use  of  such 
diagrams. 

106.  Total  Turning  Effort  T  may  be  found  by  combining  the  results  of 
(10),  (16)  or  (23),  (18)  or  (22),  and  (41);  or: 


T  —  PT  + 


(47) 


The  most  common  use  of  turning-effort  diagrams  is  in  connection  with 
the  determination  of  flywheel  weight  for  electrical  service,  in  which  the 
load  curve  is  a  straight  line.  If  it  is  desired  to  make  such  investigations 
for  compressors  of  any  type,  whether  arranged  in  tandem,  or  on  a  common 
shaft  with  different  crank  angles,  these  reversed  diagrams  must  be  plotted 


H.SUCT. 
c  COM  P. 


H.EXfi 

C.  EXH. 


H.  EXH. 
C.  5UCT. 


FIG.  193. 


with  their  inertia  diagrams  and  added  algebraically  to  the  values  found 
by  (47). 

Where  possible,  the  formulas  are  given  in  such  a  way  that  results  may 
be  obtained  either  by  measuring  the  diagram  or  by  calculation.  In  the 
latter  case,  P,  W  or  F  are  multiplied  by  some  factor  depending  upon  the 
value  of  angle  0.  To  facilitate  calculation  a  number  of  these  values  are 
given  in  Table  53  for  a  complete  revolution,  the  crank  circle  being  divided 
into  24  equal  parts. 

An  approximation  to  Formula  (30)  may  be  written: 

(48) 


The  error  is  less  than  2  per  cent,  if  n  is  4,  being  maximum  when  6  is  90 
degrees.  This  formula  is  used  for  finding  values  in  Table  53, 1  being  zero 
at  the  center  of  the  crosshead  pin.  The  minus  sign  in  the  table  indicates 
that  the  curve  has  crossed  the  plotting  line,  but  has  no  other  significance. 


THE  SLIDER  CRANK 


311 


For  a  4-cycle  internal-combustion  engine  the  values  are  all  repeated, 
but  PT,  on  the  second  revolution  is  different,  as  it  depends  upon  the  pres- 
sure on  the  piston. 

Steam  Engine  Diagram. — As  an  example  of  application,  data  will  be 
assumed  for  a  simple  steam  engine,  the  values  given  in  Table  54,  then 
plotted  in  Fig.  194.  The  indicator  and  inertia  diagrams  for  the  example 
are  given  in  Figs.  185  and  186.  Data  for  the  problem  will  be  for  the  20- 
by  48-in.  Corliss  engine  designed  in  Chap.  XII,  the  parts  of  which  are 
designed  in  later  chapters.  The  data  necessary  for  the  problem  are: 
Weight  of  reciprocating  parts  =  1550  Ib.  Weight  of  connecting  rod  = 
1330  Ib.  n  =  6,  N  =  100,  o>2  =  109.  Other  data  for  the  rod  found 
from  methods  in  Par.  99  are:  I  =  78,654,  r2  =  59,  LG  =  6.32,  LP  =  8.07. 


/ 

7 

\ 

^ 

--.», 

x 

K                                    y 

/ 

-^ 

x 

—  ^ 

[\ 

J 

/ 

\ 

0 

/ 

/ 

\ 

\ 

Of      Z  .3 


FIG.  194. 


The  weight  of  crank  referred  to  pin  =  500  Ib.  Weight  of  counter- 
balance referred  to  pin  =  2225  Ib.  The  inertia  of  the  reciprocating  parts 
are,  from  (33)  and  (34):  FH  =  12,250  Ib.,  and  Fc  =  8750  Ib.  Other 
points  in  the  curve  were  found  from  Klein's  construction,  Fig.  182.  The 
connecting  rod  is  not  included  with  the  reciprocating  parts  but  is  treated 
separately.  Total  pressure  P  includes  the  effect  of  inertia  of  the  recip- 
rocating parts  proper. 

The  counterbalance  overbalances  the  crank  by  1725  Ib.,  referred  to 
the  crank-pin  center.  The  signs  for  CT  are  then  opposite  to  those  given 
in  Table  50. 

If  the  area  of  Fig.  194  is  found  by  a  planimeter  and  divided  by  the 
length  the  mean  ordinate  may  be  found,  and  a  line  representing  the  mean 
turning  effort  drawn  as  shown.  The  fluctuation  of  turning  effort  may 
thus  be  plainly  seen,  and  the  areas  between  the  curve  and  this  line  repre- 
sent energy  fluctuations.  The  ratio  of  the  maximum  area  thus  enclosed 
to  the  area  of  the  rectangle  formed  by  the  mean  line  and  the  plotting  line 
(for  one  stroke,  one  revolution  or  one  cycle,  as  desired)  is  called  the  coef- 
ficient of  the  fluctuation  of  energy.  Referred  to  one  stroke,  this  quantity 
for  Fig.  194  is  0.333.  The  mean  line  must  always  be  found  from  the 
diagram  for  the  entire  cycle. 


312 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

o  c 


oo" 


H 

I 


gj 


o 

i 

CO 


8 


8 


8 


00 


So 


THE  SLIDER  CRANK 


313 


c-; 


•o 


o 


8 


CO 


•o" 

rH 

O 

g 

rS 
O 


0 


o" 

CO 


<o 


§ 


1 


314 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Combined  Turning-effort  Diagram. — When  a  steam  engine  has  more 
than  one  cylinder,  the  diagrams  may  be  combined.  This  is  especially 
interesting  for  twin  or  cross-compound  engines  with  cranks  not  in  the 
same  plane;  the  most  common  angle  between  2-cylinder  engines  is  90 
degrees. 


L.P. 


CYL'S.      •& 


H.R 


FIG.  195. 

While  the  crank-effort  diagrams  for  the  high-  and  low-pressure  cylinders 
of  a  compound  steam  engine  would  not  be  just  alike,  or  the  same  as  Fig. 
194,  they  may  be  assumed  the  same  and  a  combined  diagram  plotted. 
This  will  in  reality  be  for  twin  simple  engines,  and  the  combined  diagram 
will  be  the  same  whichever  crank  leads;  with  cross-compound  engines, 
however,  a  more  uniform  turning  effort  may  sometimes  be  obtained  with  a 
certain  crank  leading,  and  this  may  be  determined  by  trial. 


012 


FIG.  196. 


It  is  convenient  to  refer  all  forces  to  one  crank  circle,  such  as  the  high- 
pressure,  the  numbers  on  the  rectified  crank  circle  on  which  the  diagram 
is  plotted  referring  to  this  circle.  Thus  Fig.  195  is  a  crank  circle  for  a 
cross-compound  engine  in  which  the  low-pressure  crank  leads;  therefore 
when  the  high-pressure  crank  is  on  position  0(or  24)  of  its  diagram  (which 


THE  SLIDER  CRANK  315 

is  also  the  zero  of  the  combination  diagram),  the  low-pressure  crank  is  on 
position  6  of  its  diagram,  and  the  force  at  this  point  must  be  plotted  at 
position  0  of  the  combined  diagram.  When  the  high-pressure  crank  is 
at  18,  the  low-pressure  crank  is  at  0;  then  if  the  low-pressure  diagram 
were  drawn  on  tracing  cloth  and  moved  along  so  that  its  0  position  falls 
upon  the  18  position  of  the  combined  diagram,  the  right  relation  would  be 
established.  This  is  shown  in  Fig.  196.  If  the  ordinates  of  these  dia- 
grams are  now  added  algebraically,  the  combined  diagram  formed  by  the 
highest  curve  is  formed. 

The  mean  line  may  be  drawn  as  for  Fig.  194.  It  will  be  noticed  that 
the  energy  fluctuations  are  more  frequent  but  less  in  intensity. 

By  changing  the  angle  between  the  cranks  it  is  sometimes  possible  to 
reduce  the  maximum  fluctuation  of  energy;  the  writer's  attention  was  first 
called  to  this  in  the  paper  by  Mr.  Astrom  in  vol.  xxii,  Trans.  A.S.M.E. 
referred  to  at  the  end  of  Chap.  XVIII.  This  is  seldom  done  in  practice, 
and  unless  the  improvements  were  very  marked,  would  probably  not 
usually  be  considered  worth  while. 

An  examination  of  Figs.  185,  186  and  194  shows  quite  a  reduction  of 
the  crank-end  areas  on  account  of  the  reduced  piston  area  due  to  the 
piston  rod;  by  making  the  cut-off  for  the  crank  end  longer  than  that  of 
the  head  end,  the  work  may  be  equalized,  probably  resulting  in  a  more 
uniform  turning  effort.  This  adjustment  would  be  possible  with  Corliss 
gear  without  any  change  in  design. 

Internal-combustion  Engine  Diagram. — A  combined  indicator  and 
inertia  diagram  for  a  4-«cycle,  single-acting  internal-combustion  engine  is 
shown  in  Fig.  187.  This  was  drawn  to  scale  for  a  3J^-  by  4-in.  gasoline 
engine  running  1500  r.p.m.  and  may  be  used  as  an  illustration  for  plot- 
ting crank-effort  diagrams.  The  indicator  diagram  used  is  conventional, 
and  plotted  from  Par.  80,  Chap.  XIV,  assuming  an  absolute  compression 
pressure  of  100  lb.,  and  that  the  m.e.p.  is  88  Ib.  Modifications  were  made 
as  suggested;  the  suction  and  exhaust  lines  are  taken  as  atmospheric. 
The  connecting  rod  is  not  included  with  the  reciprocating  parts  but  is 
treated  separately.  The  total  pressure  P,  however,  includes  the  inertia 
of  the  reciprocating  parts  proper.  The  piston  is  taken  as  cast  iron. 
Other  necessary  data  are  as  follows:  Weight  of  piston  and  pin  =  2.5  lb. 
Weight  of  connecting  rod  =  2  lb.  I  =  0.636,  <o2  =  24,500,  L0  =  0.77, 
n  =  4.  The  inertia  of  the  piston  at  the  head  end  is  397  lb.  and  at  the 
crank  end  238  lb.  Table  55  is  prepared  from  these  data.  The  effect  of 
gravity  being  relatively  small,  is  neglected. 

The  turning  effort  diagram  for  a  single  cylinder  is  plotted  in  Fig.  197. 
Due  to  the  idle  strokes  during  exhaust  and  suction,  the  effort  is  negative 


316 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


00  00 

2 

£>  go 

o 

0 

0 

0 

co  oo 

iO  <M 

0 

O 

1 

1 

CO 

CO 

iO 

(N 

CO 

* 

- 

o 

CO 

CO 

7 

rH 
rH 

3 

1 

rH 

05 

O 

i 

§ 

CO 

i 

GO 

e 

1 

05 
rH 

(N 

rH 

o 

S3 

i 

. 

O5 

rH 

8 

id 
1 

O5 

1 

CO 

n 

00 

a 

1 

0 

8 

* 

co 

'oo 

O5 

<N 

00 

cS 

rH 

rH 

CO 

1 

GO 

1 

R 

^: 

05 
"* 

1 

J> 

CO 

iO 

(N 

a 

„ 

8 

id 

rH 
1 

id 

g 

S 

| 

CO 

1 

.2 

1 

'i 

CO 

co 

CO 

• 

^H 

0 

0 

^j 

00 

CO 

c5 

i 

rH 

*^* 

*•"*" 

1 

00 

CO 

rH 

t> 

iO 

•^1 

C^l 

•9 

1 

rH 
1C 
1 

0 

1 

% 

CO 

I 

00 

§ 

o 

3 

8 

00 

rH 

(N 

iO 

rH 
1 

rH 
1 

CO 

IS 

T 

rH 

i 

s 

g 

g 

s 

rH 

(N 

^ 

0 

~f 

0 

CO 

1 

7 

10 

1 

05 

s 
i 

8 

rH 

(N 

rH 

rH 

t^ 

CO 

i 

00 

O5 

IO 

oo 

N 

<N 

rH 

iO 

00 

O5 

CO 

CO 

O5 

1 

1 

1 

CO 
1 

rH 

<N 

•^ 

CO 

CO 

CO 

IO 

O 

*o 

t~} 

^^ 

o> 

»—  ( 

c^ 

CO 

iO 

rH 

O5 

J>^ 

1 

co 

1 

^f 

C^ 

C^ 

l> 

£>. 

1 

1 

I 

So 

co^ 

1 

o" 

rH 
2J. 

rH 

§ 

9) 
ft| 

H 

^ 

** 

fc, 

*** 

Ct, 

*" 

^ 

** 

^ 

h 

fc. 

THE  SLIDER  CRANK 


317 


oo  06 

06  06 

« 

S  IS 

© 

0 

0 

£> 

^  55 

9 

0 

1 

1 

8 

<c 

CO 

CO 

>0 

10 

10 

8 

i-H 

c^ 

CO 

T^i 

C^ 

TT 

T^ 

fH 

1 

cq 

• 

CO 

s 

00 

0 

10 

0 

<N 

<N 

T-l 
T-H 

»0 

s 

s 

CO 

a 

1 

£ 

O5 

T-H 

<M 

TH 

05 

1 

i 

s 

g 

s 

O5 
CO 

00 

i-H 

00 

S 

TH 

0 

CO 

00 

s 

CO 

TH 

TH 

TH 

1 

1 

T—  1 

00 

10 

s 

TH 

00 

10 

2 

s§ 

T-H 

o 

jfcj 

CD 

TH 

»o 

§* 

1 

CD 

CO 

CD 

05 

00 

Tti 

"* 

0 

0 

00 

s 

s 

1 

1 

1 

1 

1 

1 

<N 

^ 

CO 

10 

O 

(N 

£ 

CO 

t» 

00 

IO 

»o 

O5 

£>. 

iH 

TH 

1 

1 

T-H 

; 

1 

TH 

1 

1 

1 

0 

g 

T-H 
T-H 

CO 
CO 

00 

oo 

05 
00 

CO 

3 

a 

TH 

1 

I 

<N 

I 

1 

1 

1 

s 

T-H 

1 

T-H 

1 

Si 

id 

T-H 

05 

CO 

6 
co 

oo 

g 

TH 

1 

§ 

1 

1 

1 

S3 

§ 

CO 

oo 

05 

(N 

TH 

co 

TH 

JB 

C^ 

oo 

TH 

00 

O5 

TH 

I 

1 

»o 

t^ 

C<l 

c^ 

1 

1 

1 

1 

s 

s 

8 
1 

CO 

id 

TH 

CD 

TH 

05 

i 

1 

1 

1 

1 

1 

§ 

I 

I 

1 

i 

0 

T-H 

§ 

fe 

H 

h 

fe; 

Ex 

i 

s 

£ 

* 

S                               v  ' 

318 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


00    00 

« 

°°  QC 

0 

O 

1 

CO 

t- 

05 

8 

5 

3 

(N 

TH 

CO 

ct 

8 

O5 

TH 
TH 

8 

00 

§ 

§ 

CO 

i 

I 

s 

TH 

CO 

s 

TH 

s 

.2 

1 

8 

0^ 

00 

s 

? 

0 

g 

TH 

oo 

"5 

a 

1 

1 

1 

00 

oo 

TH 

i 

<N 

1 

1 

1 

0 

TH 

TH 

(M 

1 

1 

1 

»o 

8 

s 

1 

1 

1 

s 

3 

§ 

TH 
TH 

1 

1 

£ 

o" 

p 

0 

iH 

N  

**—s 

i 

s 

£ 

^ 

00    00 

00   00 

O 

0 

9 

1 

ft 

0 
CO 

i 

b» 

1 

TH 

^H 

* 

1 

1 

5 

00 

3 

3 

TH 

1 

1 

1 

TH 

5 

CO 

3 

10 

TH 

TH 

I 

1 

1 

CO 

£ 

£ 

^ 

TH 

1 

* 

1 

1 

9 

rH 
1 

TH 

TH 

| 

TH 
TH 

1 

9 

1 

TH 

I 

TH 

1 

•<* 

7 

7 

1 

fe 

0 

g 

5 

7 

TH 

1 

1 

5 

C<l 

g 

i 

1 

1 

00 
CO 

to 

8 

i 

1 

1—  1 

TH 

1 

CO 

CO 

8 

1 

8 

a 

^ 

/-x 

1 

TH 

s 

8 

IN 

i 

Qs 

THE  SLIDER  CRANK 


319 


during  the  first  part  of  the  stroke,  inertia  only  having  any  effect  (if 
suction  and  exhaust  are  considered  as  atmospheric). 

The  mean  line  may  be  drawn  as  described  under  steam  diagrams,  and 
the  fluctuation  of  energy  determined;  the  maximum  value  of  &E  (Chap. 
XVIII)  is  obvious. 


Combined  Turning-effort  Diagrams. — Fig.  197  may  be  considered  a 
diagram  for  a  single-cylinder  engine,  or  cylinder  No.  1  of  a  multi-cylinder 
engine.  Some  of  the  most  common  arrangements  will  now  be  given, 
the  diagrams  for  the  different  cylinders  being  first  shown  separately  to 
avoid  confusion,  and  then  combined. 


Ate/, 


no" 


FIG.  198. — Two-cylinder  diagrams. 

Twin  cylinder  engine  with  cranks  in  unison.  Order  of  firing,  1-2. 
In  the  crank  diagram  of  Fig.  198,  the  head-end  dead  center  is  assumed  to 
be  at  the  top  of  the  circle,  as  in  a  vertical  engine.  The  turning-effort 
diagram  is  dimensioned  in  degrees,  starting  from  the  head-end  dead  center 
of  cylinder  No.  1.  The  combined  diagram  is  given  below  that  of  cylinder 


320 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


No.  2.  The  crank  is  assumed  to  turn  clockwise,  but  to  follow  the  order  of 
firing,  the  crank  circle  should  be  followed  in  a  counter-clockwise  direction. 

Three-cylinder  engine  with  cranks  at  120  degrees.  Order  of  firing, 
1-3-2.  See  Fig.  199. 

Four-cylinder  engine  with  cranks  at  180  degrees.  Order  of  firing, 
1-3-4-2.  See  Fig.  200. 

Six-cylinder  engine  with  cranks  at  120  degrees.  Order  of  firing, 
1-5-3-6-2-4.  See  Fig.  201. 


No.  I 


FIG.   199. — Three-cylinder  diagrams. 

The  cranks  are  differently  arranged  on  some  shafts  as  shown  in  Fig. 
413,  Chap.  XXVIII,  for  which  the  firing  order  is  1-2-4-6-3-5. 

Crank  diagrams  for  8-cylinder  and  12-cylinder  engines  are  shown  in 
Figs.  202  and  203  without  the  crank  effort  diagrams.  These  may  be 
plotted  in  the  same  way  when  the  order  of  firing  is  known;  this  will  be 
considered  in  Chap.  XX.  The  dotted  lines  of  the  crank  diagrams  show 
equivalent  positions  of  the  cranks  to  give  the  same  turning  effort  if  the 
cylinders  were  all  vertical. 


THE  SLIDER  CRANK 


321 


It  will  be  seen  that  in  all  cases  the  order  of  firing  is  such  as  to  divide 
the  impulses  around  the  circle  evenly;  aside  from  this  there  is  not  a  fixed 
standard,  but  the  more  usual  timing  is  given  in  this  chapter. 

107.  Reactions  on  the  Frame. — With  the  exception  of  the  connecting 
rod  in  vertical  and  angle  engines,  gravity  of  the  moving  parts  always 
exerts  a  downward  effect  only.  The  magnitude  of  the  normal  component 


f-4 


2-3 


180° 


S401 


720  9 


No.  2 


No.4 


\\ 


\/ 


\/ 


\ 


FIG.  200. — Four-cylinder  diagrams. 

of  gravity  is  given  by  (19),  and  it  always  acts  toward  the  center  line  of 
the  engine  at  the  crosshead  end  when  the  cylinder  is  above  the  crank, 
and  away  from  the  engine  at  the  crank.  The  weight  of  the  reciprocating 
parts  of  a  vertical  engine  is  included  in  P,  the  total  piston  pressure.  In 
horizontal  engines  the  gravity  of  these  parts  have  no  effect  except  to 

produce  some  pressure  on  the  cylinder  walls  and  guide.     Under  certain 
21 


322 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


unusual  conditions  their  changing  position  may  have  a  slight  effect  upon 
balance. 

Forces  due  to  gas  or  steam  pressure  and  to  acceleration  are  indepen- 
dent of  the  position  of  the  engine,  and  their  effect  upon  the  frame  will  be 


/Vo./f 
0 


No.2- 


120°  240°  360*  480°  600°^  720° 


No.3< 


No.  4 


No.S 


MO.  6 


\l 


\/ 


\/ 


\/ 


FIG.  201. — Six-cylinder  diagrams. 

given  here.  Force  P  in  this  paragraph  had  best  be  taken  for  steam  or  gas 
pressure  only  (including  weight  of  reciprocating  parts  for  vertical  or 
angle  engines),  the  inertia  of  the  reciprocating  parts  being  computed 
separately,  as  all  forces  except  P  (and  consequently  PN)  vary  directly  as 


THE  SLIDER  CRANK 


323 


2-2,  J-J, 

FIG.  202.— Eight-cylinder  diagram. 


4-4, 


2-5 


'3-4 


2rS, 


2-2,  3-3,          4-4-, 

FIG.  203.— Twelve-cylinder  diagram. 


6-6, 


324 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


co2;  these  may  then  be  combined  and  plotted,  and  the  effect  of  different 
rotative  speeds  easily  determined  by  changing  the  scale.  The  reactions 
due  to  P  may  be  added  to  any  of  these  inertia  diagrams  and  the  total 
effect  determined. 

Force  P  has  an  unbalanced  effect  upon  the  frame  only  through  its 
normal  component  PN.  The  normal  component  of  FP  may  be  found  from 
(43)  by  taking  LG  =  0;  then: 


F>K=FP 


1 


tein  0' 


(49) 


The  signs  for  P  assume  the  force  to  be  acting  in  the  direction  of  piston 
motion;  when  the  reverse  is  true  the  signs  are  reversed.  This  may  be 
determined  from  diagrams  similar  to  those  given  in  Figs.  186  and  187. 

Fig.  204  shows  the  numbering  of  the  sectors,  the  meaning  of  the  signs, 
and  the  action  of  the  different  forces,  while  Table  56  gives  the  signs. 

TABLE  56 


Force 

Formula 

Sector  of  circle 

1 

2 

3 

4 

5 

6 

f  PN 

(12) 





__ 





__ 

"DA 

FPN 

(49) 

+ 

— 

— 

+ 

+ 

- 

PA 

FXN 

(43) 

+ 

— 

— 

+ 

+ 

— 

1  FYA 

(42) 

+ 

+ 

+ 

— 

— 

— 

PN 

(12) 

+ 

+ 

-t- 

+ 

+ 

,+ 

FPN 

(49) 

— 

+ 

+ 

— 

— 

+ 

PBY 

'    FXN 

(43) 

— 

+ 

+ 

— 

— 

+ 

FYB 

(39) 

+ 

+ 

+ 

— 

— 

— 

FCY 

(50) 

+ 

+ 

+ 

— 

— 

- 

r  FP 

(32) 

— 

+ 

+ 

+ 

+ 

— 

PBX 

[FX 

(32) 

- 

+ 

+ 

+ 

+ 

- 

1  Fcx 

(51) 

— 

— 

+ 

+ 

— 

— 

If  the  revolving  parts  are  not  balanced,  or  are  overbalanced  due  to  a 
portion  of  the  reciprocating  parts  being  balanced  at  the  crank,  the  cen- 
trifugal force  of  this  weight  will  react  at  the  main  bearing.  Normal  to 
the  line  of  stroke  this  force  is  : 


FCY  =  S  (^  -)  flco2  sin  e 
\H    l 


(50) 


This  forms  a  part  of  pBY,  the  sign  being  as  in  Table  56  if  the  crank  and  pin 


THE  SLIDER  CRANK  325 

are  under-balanced  ;  if  overbalanced  the  signs  are  reversed.     The  effect 
parallel  to  line  of  stroke  is: 


This  is  added  to  pBX,  the  sign  being  as  in  Table  56  for  the  unbalanced 
crank  and  pin. 

The  forces  of  Fig.  204  and  Table  56  may  be  plotted  or  tabulated  sepa- 
rately, similar  to  Table  57,  and  combined  if  desired.  They  do  not  all  act 
in  the  same  plane,  and  may  produce  other  forces  and  couples.  Their 
effect  may  be  taken  up  within  the  frame  itself  in  some  cases,  so  as  to  have 
little  or  no  effect  upon  the  foundation  or  other  support,  especially  in 
multi-cylindered  engines.  Some  of  the  forces  are  insignificant  in  some 
cases,  but  with  different  types,  sizes,  speeds,  etc.,  it  is  well  to  prove  this 
by  calculation;  the  formulas  are  simple  and  easily  applied  with  the  aid 
of  Table  53  and  the  diagrams,  and  their  use  may  account  for  various  things 
giving  trouble  to  engine  builders. 

The  forcesFx  (due  to  the  connecting  rod)  andFp(due  to  the  reciprocat- 
ing parts)  act  along,  or  parallel  to  the  line  of  stroke,  but  may  act  on  the 
frame  either  at  the  cylinder  end  or  the  bearing  end,  or  may  be  divided 
between  them,  depending  upon  the  steam  or  gas  pressure  acting  upon 
the  piston.  It  must  be  borne  in  mind,  however,  that  if  the  restraint  of 
the  foundation  bolts  is  neglected,  the  forces  acting  upon  the  frame  in  the 
line  of  stroke  are  limited  by  the  steam  or  gas  pressure  in  the  cylinder; 
this  must  be  determined  by  the  indicator  diagram  alone,  inertia  having 
no  effect.  If  inertia  is  great,  part  of  it  may  be  balanced  by  compression, 
in  which  case  it  is  transmitted  to  the  foundation  from  the  cylinder  end  of 
the  engine;  any  excess  inertia  is  transmitted  to  the  frame  and  thus  to  the 
foundation  at  the  shaft  end  of  the  engine.  The  inertia  effects  normal  to 
the  line  of  stroke  cause  bending  and  torsional  stresses  in  the  frame,  but 
these  are  influenced  by  the  attachment  to  the  foundation. 

Divisions  2  and  5  in  Fig.  204  are  not  quite  the  same  for  FP  and  Fx,  as 
the  inertia  curves  do  not  cross  the  line  at  the  same  point. 

The  action  of  the  forces  pBx  and  pBY  along  the  crank  in  a  line  from  the 
center  of  the  shaft  to  the  center  of  the  crank  pin  is: 

PR  =  PBX  COS  0  +  PBY  sin  6  (52) 

All  or  part  of  the  forces  of  which  these  are  composed  may  be  taken  in 
finding  pR.     Formula  (52)  is  convenient  as  practically  all  balancing  is 
done  along  the  crank  center  line. 
The  forces  of  Fig.  204,  added  algebraically  are  : 


326 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


PA  =  PN  +  FPN  +  FXN  +  FYA 

PBX  =  FP  +  Fx  +  Fcx 

PBY  =  PN  +  FPN  +  FXN  +  ^V*  -f 


(53) 
(54) 
(55) 


Formula  numbers  for  the  various  forces  are  given  in  Table  56.  It  may 
be  convenient  at  this  place  to  state  the  meaning  of  the  notation  contained 
in  Formulas  (53)  to  (55). 


FP 
FP 
Fx 


3    /Oil , 


=  normal  component  of  force  P  due  to  steam  or  gas  pressure.  If 
P  includes  the  inertia  of  the  reciprocating  parts,  PN  includes 
FPN.  Should  the  inertia  of  the  connecting  rod  be  included  as  in 
rough  calculations.  PN  includes  FXN- 

=  the  inertia  of  the  reciprocating  parts. 

=  the  normal  component  of  FP. 

=  the  inertia  of  the  connecting  rod  parallel  to  line  of  stroke,  as- 
sumed as  concentrated  at  mass  center  of  rod. 


FIG.  206. 

FXN  =  normal  component  of  F*. 

FYA    —  effect  at  crosshead  pin  of  inertia  of  connecting  rod  normal  to 

line  of  stroke. 
FYB    =  same  at  crank  pin. 
Fcx    =  effect  of  centrifugal  force  of  crank  or  counterbalance  parallel  to 

line  of  stroke. 
FCY    =  same  normal  to  line  of  stroke. 

In  vertical  engines  GN,  the  gravity  effect  of  the  connecting  rod,  should 


THE  SLIDER  CRANK 


327 


be  added  to  pA  and  PBY',  its  value  is  given  by  (19)  and  the  signs  are  the 
same  as  for  PN  in  Table  56. 


\  XI 


X 


X 


\ 


\ 


\ 


XJ 


FIG.  207. 


With  the  data  for  the  Corliss  engine  already  considered,  and  by  the  use 
of  Fig.  204  and  Table  56,  Table  57  has  been  computed.     The  method  of 


determining  the  counterbalance  will  be  given  in  the  next  chapter.     The 
values  of  pA  are  plotted  along  the  crosshead  path  in  Fig.  205.     Values 


328 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


o  o  o  o  o 


t^    ~H  »0  ^  00 

I-H    OO  O  (O  CO 

CO    CO  ^  ^  (M 

1  1 


I      I 


CO  O  CO   O  CO 
»O    OS    CO    U5    CO 

^   I     I   ,4'f 


o  oo  o  o  oo 

rH    (N    CO    •*    N 


S  CO  §    §    § 

OS  CO  |     OO    TfH 

N  |  '  .  -5   rH 

1  I 


O  00  CO 
00  •#  (N 
r-l  1C  N 


O  O  CO  CN  »O 
00  N  C<l  O  CO 
W  00  CO  OS  N 


CO  1C  •-!  t^ 
OS  O  OS  CO 
CO  »0  .-I  •* 


§Q  O  O  o 

O  00  OO    OO 

CO  CO  ^O    >O 

00"  00"  CO"  00 


COCO'HOSOOCOOOrH<NIN'-l 


(N    00   00"  CO*  0   0 


t>»OOO 
t^-OS'O 
COoOCO 


O>OO 
OiOOO 


CDOCOOO<NO»CO»OO 
OSTjtCOOOCOOOtCCOOSO'O 


Ooooooooocot^-oo 

<-H(NCO'Ot>-OOiO»Ot>'00'-i 
(N  t>-  M  «5  <N  O3  Tt<  OS  00  (N  10 
W  CO"  CO"  OT  TH  IN  OO"  (N 


- 

(NCOOOOOOCOOrHCOCO 
OS    CO  «3    CO    CO    OJ    t>  O    CO 

<N"  co"  co"  o"  rn  ei  o 


CO       |         |      CO"    CO"    O       I      i-H       |      CO"    rH 


«S      |       |     CO    (N    C5    CO*  •*"  rH    05"  CO" 

"•  1   1   1   1  ^ 


ooooooo^oo 

iO<NCOOOt^Ot>-O 


I     I   w"  IN"  oo"  co"  co"  <N"  10"  co" 

i  1-1.7- 


OCOOOt^OOOOO 
lNiN-^OSCOiOt^COiO'* 
OOCOOOOcO_t»i-HO500O5 

\     \    •-"  «H  ic"  oT  oo"  <N"  o  o 

1  i  1  " 


iOr-lOSOOOOt^COCMCSIlN 

'  '       N"  d  f  7  a  s 
1  '  '  1 


CO   •* 

1 


S| 

o  o 

I    I 


^  « 

CO   l> 

^  1 


11 

o"  o" 

i-H    r-l 
I 


<N    OS    CO    <N 


<N    CO 

T}f      IO 


. 

0,  b,  b,  &,    A 


fe,  fe,  fe,  &, 


68 

fc,  fe, 


THE  SLIDER  CRANK 


329 


o  o  o  o  o 


2  °  »  2 

•*  10  •-1  •* 

I   I   I   I 


OS  O  CO  IN  TJ< 

CO  <N  <N  O  00 

<N  CO  CO  OS  <N 

I  I  I  I  <N 


oooooo^- 


O  V5  O  1C  iO 
00  IN  00  00  00 

CN  r^  co  co  co 


(N  OS  CO  10  10 

7-.i-i.Jf" 


o  ^ 

10* 


OOoOO 
^OSiOiO 


'-H'-'INOSOOlN 

1    1    1    1    1    1 


8§£8 

CO    CO    N    CO 

1  '  f  7 


t*  00  1-1  O  to 

t>«  ^  CO  M4  OS 

CO  i-H  i-H  t^  CO 

<N  |  |  -J  V 

\  1  1 


O   CO    00   0* 

CO    lO    O    •**! 
CO  CO   Is- 

i4  TjT 

1    1 


§00  O  O  M 

cxi  co  •*  co 

•*  rH    1O 

I         I   I 


o  o  to  o  *# 

a.  ^  ro  to  ^. 

-*  OS   CO   «5  t^ 

>o            —  — 

I           I  I 


(N    •* 
I       I 


"N    t» 
O   IN 

os  q 

I    « 

I 


oo  to 
co  -H 


IN    OS   CO   IN    CO 


to_  •*  oo  os  co  •*  to  o 

M"  (N    N    CO*  tO*  <N    10    CO 
I       I       I       I       I       I     ?    ^ 


OoOtOOOcOOOOO 

i  i  i  i  i  7  T 


i.      _ 
N  CO*  CO    TjH*      |      rH       |      CO    TlT 

III      1      1 


COCO      |tOCOi-<O>t^  CO 

CO      |  CO    CO    T(T  1-1  fff 

1    1    1 


S  8  §  S  S  8  §  S  fc  i  g 


CO   CO   CO 

1   1   1 


IO"      I        I      CO    <N"    -H    CO"   •*"    rH    00    00* 

III 


COOiO 


CO     |      |     ,_,    r-t   I-H   00*  t>T  of  00* 

III* 


00*  00   CO* 


II 

o  o 


II 

•*"  co 


88 


0  0 

1 


a,  a, 


fc,  ft. 


330 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


of  pBY  and  FCY  (for  counterbalance)  are  plotted  on  a  rectified  crank 
circle  in  Fig.  206,  and  of  pBX  and  FCx  in  Fig.  207.  The  resultant  ps  of 
forces  PBX  and  pBv  may  best  be  found  graphically  from  Figs.  206  and  207. 
This  has  been  done,  taking  the  counterbalance  into  consideration,  and 
plotted  on  a  crank  circle  in  Fig.  208,  giving  both  direction  and  magnitude 
of  the  force  acting  during  the  revolution  due  to  the  moving  parts  of  the 


FIG.  209. 

slider-crank  mechanism.  This  force  acts  on  the  main  bearing.  To  ob- 
tain the  force  acting  at  the  crank  pin,  that  due  to  the  crank  and  counter- 
balance (FCY  and  FCX)  must  be  omitted.  This  is  shown  in  Fig.  209. 
The  actual  force  on  the  bearing  is  somewhat  different  as  it  is  not  in  the 
same  plane;  neither  do  the  forces  due  to  crank  and  counterbalance  act 
in  the  same  plane  as  the  forces  at  the  center  of  the  crank  pin,  but  for  most 
calculations  for  strength  and  wear  this  may  be  neglected. 


FIG.  210. 

To  obtain  the  total  force  acting  on  the  crank  pin  or  shaft  at  any  point, 
the  resultant  of  ps  and  the  steam  or  gas  pressure  must  be  obtained;  when 
bearing  pressure  is  desired  the  effect  of  gravity  must  be  included  (con- 
necting rod,  crank,  shaft,  wheel,  etc.).  The  force  due  to  steam  or  gas 
pressure  may  be  found  by  the  use  of  a  stroke  diagram  such  as  Fig. 
186  or  187,  neglecting  inertia.  In  Fig.  210  it  may  be  seen  that  the  force 


THE  SLIDER  CRANK 


331 


PR  acts  at  an  angle  0  -\-  <j>  =  a,  from  the  point  on  the  pin  which  is  nearest 
the  cylinder  when  the  crank  is  on  the  head-end  center,  and  that  it  moves 
around  the  pin  in  a  counter-clockwise  direction.  This  is  also  true  of  the 
shaft. 


FIG.  211. 


Fig.  208  or  209  may  be  combined  with  Fig.  210;  then  to  find  the  forces 
acting  on  shaft  or  pin,  these  may  be  drawn  in  the  position  of  head-end 
dead  center;  by  revolving  the  resultant  diagram  on  the  same  center  in  a 
counter-clockwise  direction,  the  resultant  forces  may  be  transferred  to 


FIG.  213. 


FIG.  214. 


the  pin  or  shaft  in  such  a  manner  that  they  pass  through  the  center  if 
produced.  Figs.  209  and  210  for  the  crank  pin  are  combined  in  Fig. 
211  and  transferred  to  the  pin  in  Fig.  212.  The  latter  shows  the  normal 
force  acting  on  all  sides  of  the  crank  pin  by  means  of  which  the  maximum 
reversed  or  repeated  loads  may  be  determined. 


332  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Figs.  208  and  210  for  the  shaft  are  combined  in  Figs.  213  and  214  in 
the  same  way. 

These  diagrams  are  for  the  20  by  48  in.  Corliss  engine  previously  men- 
tioned, and  it  must  be  remembered  that  they  are  not  strictly  accurate  due 
to  the  forces  not  being  in  the  same  plane.  In  all  of  these  diagrams  except 
Figs.  212  and  214,  the  force  acts  from  the  point  on  the  circle  from  which 
the  line  starts,  and  the  effect  of  gravity  of  the  parts  is  neglected.  In 
Figs.  212  and  214  the  force  curve  is  all  drawn  outside  the  circle  which 
represents  pin  or  shaft,  and  acts  toward  the  circle;  the  radial  force  lines 
are  numbered  in  order  showing  the  direction  of  force  change  during  the 
cycle.  Reversed  and  repeated  loads  may  be  taken  from  these  digarams 
(Figs.  212  and  214),  the  reversed  loads  being  diametrically  opposite.  If 
two  forces  are  more  than  90  degrees  apart  they  produce  a  certain  amount 
of  reversal;  this  usually  produces  bending.  As  the  stress  measured  from 
any  neutral  axis  is  proportional  to  the  distance  from  the  axis,  the  inten- 
sity and  kind  of  stress  may  be  found  at  any  point  on  the  surface  for  forces 
inclined  at  any  angle. 

Various  couples  are  produced  by  the  constituents  of  pA  and  PBY, 
some  of  which  tend  to  neutralize  the  others.  Their  effect  is  also  influenced 
by  the  counterbalance  if  any  is  used. 

The  attachment  of  the  frame  to  the  foundation  or  other  support  in- 
fluences the  effect  of  these  forces  which  might  be  determined  by  taking 
moments  about  certain  points;  but  the  origin  of  moments  is  so  indefinite, 
and  the  behavior  of  the  supports  under  active  forces  so  uncertain  that 
but  little  satisfaction  is  obtained  by  an  attempt  at  anything  like  accurate 
analysis.  For  large  engines,  as  careful  a  consideration  as  possible  of  the 
proportion  of  the  reciprocating  parts  to  be  balanced,  and  for  small  light 
engines,  a  careful  application  of  the  best  balancing  methods,  will  bring 
the  most  satisfactory  results. 

In  all  that  precedes,  an  absolutely  uniform  motion  of  the  crank  pin  has 
been  assumed.  This  is  not  strictly  the  case,  but  in  most  cases  there  is  no 
great  error  in  the  assumption,  and  an  attempt  to  account  for  this  would 
be  useless. 

The  treatment  of  the  mechanics  of  the  slider  crank  given  here,  while 
simple  in  principle,  is  rather  complicated,  and  for  most  design  problems 
only  the  direct  forces  due  to  gas  or  steam  pressure  and  to  heavy  weights 
such  as  the  flywheel,  are  used.  However,  no  one  is  absolutely  sure  of  the 
wisdom  of  neglecting  certain  forces  until  he  has  worked  through  a  number 
of  problems,  and  for  turning  effort  diagrams  used  in  flywheel  design,  the 
neglect  of  the  effect  of  acceleration  results  in  a  method  but  little  short  of  a 
rough  rule  of  thumb.  Also,  when  inertia  is  combined  with  the  stroke 


THE  SLIDER  CRANK  333 

diagram  of  Fig.  187,  the  resulting  force  diagram  differs  considerably  from 
the  original  diagram  due  to  gas  pressure  only;  were  the  connecting  rod 
included,  giving  the  forces  acting  on  the  crank  pin,  the  difference  would 
be  still  greater. 

Perfect  balance  is  claimed  for  certain  engines  of  ordinary  construction ; 
there  is  no  such  thing,  and  the  statement  either  shows  lack  of  knowledge 
or  an  attempt  at  a  "snappy"  piece  of  advertising. 

The  heat  engine  designer  cannot  afford  to  be  ignorant  of  the  simple 
principles  of  this  chapter  even  though  he  uses  them  but  little.  If  the 
forces  are  carefully  calculated  the  factor  of  judgment  may  be  reduced  in 
selecting  the  factor  of  safety;  if,  as  is  usually  the  case,  it  is  considered 
unnecessary  to  make  such  elaborate  calculations,  the  factor  of  judg- 
ment should  be  such  as  to  cover  discrepancies.  This  will  be  further 
discussed  in  Par.  166,  Chap.  XXI  and  in  connection  with  the  various  details 
in  later  chapters. 

Reference 
Balancing  of  engines W.  E.  Dalby. 


CHAPTER  XVII 
BALANCING 

108.  Introduction. — The  subject  of  the  balancing  of  reciprocating 
engines  is  well  covered  by  Prof.  W.  E.  Dalby  in  his  excellent  work  by  this 
title,  and  it  is  not  possible  to  claim  a  complete  treatment  in  a  short  chap- 
ter; however,  an  attempt  is  made  to  bring  the  essentials  of  balancing  into 
simple  practical  form,  so  that  they  may  be  used  to  solve  all  balancing 
problems  connected  with  the  usual  forms  of  reciprocating  engines.  It 
might  be  more  correct  to  say  that  the  lack  of  balance  may  be  determined 
by  these  principles  with  a  view  to  keeping  it  a  minimum. 

Notation. 

W  =  weight  in  general  in  pounds. 

WB  =  weight  to  be  balanced  referred  to  the  crank  pin. 

WP  =  weight  of  reciprocating  parts  in  pounds. 

W  =  unit  weight  in  pounds. 

C  =  weight  of  crank  and  pin  in  pounds  referred  to  pin  center. 

P  =  force  in  general  in  pounds. 
Cp  =  radial  inertia  (centrifugal  force)  of  a  revolving  weight  in  pounds. 

Q  =  a  couple. 

M  =  moment  in  pound -feet  or  pound-inches  of  weights  to  be  balanced. 
AB  =  area  in  square  inches  or  square  feet  of  face  of  counterweight. 
t  =  thickness  of  "counterweight  in  inches  or  feet. 
I  =  dimensions  in  general  in  inches  or  feet. 
r  =  radii  in  general. 

L  =  length  of  connecting  rod  from  center  to  center  in  feet. 

R  =  crank  radius  in  feet. 

w  =  angular  velocity  in  radians  per  second. 

N  =  r.p.m. 

n  =  L/R. 

g  =  32.16. 

109.  Simple,  or  Primary  Balancing.— Centrifugal  force  tends  to  dis- 
place a  revolving  weight  in  a  radial  direction  from  the  axis  about  which 

334 


BALANCING 


335 


it  revolves.     The  magnitude  of  this  force  is  equal  to  the  product  of  its 
mass  and  radial  acceleration;  or: 

W  W       /TrN\  2 

r*                E>   9  z?  /        i  / 1  \ 

OF   = /vo>2  = ti  I  -JTJT- 1  (1) 

Had  W  been  taken  as  an  indefinitely  small  division  of  the  weight  and 
R  its  distance  from  the  axis,  the  sum  of  these  products  would  equal  the 
total  weight  multiplied  by  the  distance  of  its  center  of  gravity  from  the 
axis.  It  then  follows  that  two  weights  diametrically  opposite,  revolving 
about  an  axis,  will  be  in  balance  if  the  product  WR  is  the  same  for  each, 


FIG.  215. 

and  this  is  the  general  principle  of  practical  counterbalancing  of  machine 
parts. 

The  balancing  of  parts  revolving  in  a  circle  is  simple,  and  practically 
consists  in  suitably  placing  the  balance  weights  so  as  to  make  a  neat 
design.  However,  for  over-hung  cranks  there  is  an  unbalanced  couple 
due  to  the  impossibility  of  placing  the  counterbalance  in  the  same  plane 
as  the  portion  of  the  weight  to  be  balanced.  This  is  shown  in  Fig.  215. 

So  far  as  forces  are  concerned,  balance  is  obtained  if  the  connecting 
rod  is  disconnected  when: 

=  Will  +  WR  (2) 


336  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

where  W  is  the  weight  of  the  portion  of  the  pin  in  the  bearing,  Wi  the 
weight  of  the  crank  and  W  2  the  weight  of  the  counterbalance.  The  crank 
and  counterbalance,  as  shown,  are  in  the  same  plane,  but  the  pin  is  not 
in  the  plane  with  the  portion  of  the  weight  which  balances  it,  and  the 
couple  : 

W 

—  RuPm 

g 

is  formed,  which  has  a  tendency  to  cause  rotation  in  a  plane  normal  to  the 
plane  of  rotation.  In  practice  this  is  usually  small  relatively  and  is  ig- 
nored. With  the  center-crank  engine  the  counterbalance  is  divided 
between  the  two  crank  arms  and  the  couples  balance. 

Reciprocating  Parts.  —  Let  the  effect  of  angularity  of  the  connecting 
rod  be  temporarily  neglected  and  its  weight  assumed  added  to  the  recip- 
rocating parts.  These  parts  must  be  brought*to  rest  at  the  end  of  each 
stroke,  the  force  required  being  the  product  of  their  mass  and  the  radial 
acceleration  of  the  crank.  If  C  be  the  weight  of  crank  and  pin  referred 
to  the  center  of  the  crank  pin,  and  Wp  the  weight  of  the  reciprocating 
parts,  the  force  acting  at  the  crank  pin  at  dead  center  is: 


g 

If  the  crank  and  pin  only  are  balanced,  then  at  the  dead  center,  Wp  is 
unbalanced,  and  this  is  true  in  a  lesser  degree  for  every  other  position  of 
the  crank  except  when  it  is  at  right  angles  to  the  line  of  stroke.  If  WP 
is  also  balanced  by  a  revolving  weight,  the  system  is  in  balance  only  at  the 
dead  centers,  the  counterbalance  being  too  heavy  by  the  amount  WP 
when  the  crank  is  at  right  angles  to  the  line  of  stroke.  A  compromise  is 
usually  made  by  balancing  all  of  the  revolving  parts  and  a  certain  per- 
centage of  the  reciprocating  parts;  at  dead  center  the  engine  is  under- 
balanced  and  at  right  angles,  overbalanced. 

The  percentage  of  reciprocating  parts  to  be  balanced  depends  upon  the 
type  of  engine  and  location.  For  locomotives,  a  special  formula  is  used 
in  which  the  total  engine  weight  is  a  factor,  but  the  limits  are  usually 
between  55  and  65  per  cent. 

In  all  engines  with  but  one  rod  connecting  with  the  crank  pin  this 
state  of  unbalance  must  exist,  even  though  the  system  as  a  whole  may  be 
balanced  in  a  multi-cylinder  engine. 

In  many  small  engines  with  slow  or  moderate  speed  and  heavy  frames, 
the  counterbalance  is  omitted,  or  if  added  by  the  use  of  a  disc  crank,  no 
calculations  are  made  in  proportioning  it  ;  but  with  large  engines  or  high 


BALANCING  337 

speeds,  counterbalance  is  of  advantage,  not  only  for  smoothness  of  opera- 
tion but  for  relieving  undue  wear  and  strain. 

110.  Secondary  Balance. — If,  in  addition  to  balancing  the  revolving 
parts,  a  weight  be  placed  opposite  the  crank  so  that  the  product  of  weight 
and  distance  to  mass  center  equals  the  product  of  crank  radius  and  weight 
of  reciprocating  parts,  the  radial  inertia  of  this  weight  at  any  point  in  its 
path  is  equal  to  that  given  by  (1).  From  (33)  and  (34),  Chap.  XVI,  the 
radial  inertia  of  the  reciprocating  parts  is  greater  than  this  at  the  head 
end  of  the  stroke  and  less  at  the  crank  end.  If  the  rod  were  infinitely 
long,  or  a  Scotch  yoke  were  used,  there  would  be  perfect  balance  at  the 
dead  centers,  and  this  is  known  as  primary  balance.  The  balance  along 
the  line  of  stroke  which  would  be  necessary  to  compensate  for  rod  angu- 
larity is  known  as  secondary  balance.  It  does  not  include  the  inertia  of 
the  rod  itself,  but  only  such  masses  as  may  be  considered  concentrated 
at  the  crosshead  pin. 

From  (32)  and  (48)  of  Chap.  XVI,  taking  I  as  zero  in  the  latter,  the 
inertia  of  the  reciprocating  parts  along  the  line  of  stroke  is: 

tffcos  6  +  y  cos  26  1  (3) 

L  L          j 

This  is  the  approximate  formula,  but  is  sufficiently  accurate  for  the  pres- 
ent purpose.  The  negative  signs  are  omitted,  but  cos  8  and  cos  20  have 
signs  as  explained  in  connection  with  Formula  (30),  Chap.  XVI. 

Removing  the  brackets  and  multiplying  and  dividing  the  term  con- 
taining cos  26  by  4,  (3)  may  be  written : 

W  W  J? 

F  =  —  •  u*R  cos  0  +  —  (2w)«-  jt-  R  cos  26 
y  y  rtL/ 

=  primary  force  +  secondary  force  (4) 

This  shows  the  following: 

1.  The  primary  force  along  the  line  of  stroke  is  equivalent  to  that  which 
would  be  produced  if  the  reciprocating  parts  were  concentrated  at  the 
crank-pin  center  and  revolved  at  the  same  speed  as  the  crank. 

2.  The  secondary  force  along  the  line  of  stroke  is  equivalent  to  that 
produced  by  a  mass  equal  to  the  reciprocating  parts  concentrated  at  a 
radius  equal  to  the  fraction  R/4L  of  the  crank  radius,  and  whose  rotative 
speed  is  twice  that  of  the  crank.     When  the  crank  is  at  either  dead  center, 
this  mass  is  always  at  the  head-end  dead  center. 

The  relation  of  the  crank  with  this  imaginary  crank  is  shown  in  Fig. 
216. 

A  graphical  expression  of  (3)  is  given  in  Fig.  217,  the  radius  of  the 
imaginary  crank  in  this  case  being  R/n.  The  algebraic  sum  of  the  dis- 


338 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


tances  from  the  vertical  center  line  to  the  projections  of  the  two  cranks  on 
the  horizontal  center  line  gives  the  inertia  to  the  same  scale  that  R  gives 
the  primary  force  at  dead  center. 

It  is  obvious  that  for  engines  with  a  single  cylinder,  the  balancing  of 
the  secondary  force  is  impracticable,  therefore  the  reciprocating  parts  do 
not  permit  of  perfect  balance  with  revolving  weights. 


FIG.  216. 

Complete  balance  of  the  reciprocating  parts  by  a  revolving  weight  is 
usually  undesirable,  so  that  the  importance  of  secondary  balance  is  only 
seen  in  certain  combinations  of  multi-cylinder  engines. 

It  should  not  be  inferred  from  what  has  preceded  that  revolving  weights 
at  crank  pin  and  imaginary  crank-pin  centers  would  produce  the  same 
forces  in  all  directions  as  those  produced  by  the  reciprocating  parts;  the 


FIG.  217. 

effect  is  identical  only  along  the  line  of  stroke,  the  reciprocating  parts 
having  no  other  influence  upon  the  frame  except  through  the  normal 
component  FPN,  given  by  (49),  Chap.  XVI. 

111.  Multi-cylinder  Engines. — In  this  paragraph  a  number  of  com- 
mon arrangements  will  be  discussed  relative  to  the  balancing  of  revolving 
parts  and  purely  reciprocating  parts  along  the  line  of  stroke,  leaving  the 


BALANCING 


339 


connecting  rod  and  the  normal  components  of  cylinder  pressure  and 
inertia  to  later  paragraphs. 

Complete  balance  includes  the  balance  of  forces  and  couples.     Un- 
balanced forces  may  be  determined  by  taking  the  algebraic  sum  of  the 
components  of  all  forces  normal  to  a  given 
diameter,    assuming   them   to   act  in   the 
same  plane  normal  to  the  axis  of  revolu- 
tion,  as  in  Fig.  218.     Perfect  balance  is 
obtained  if  Pi  +  P2  +  Pa  =  0. 

Unbalanced  axial  couples  may  be  deter- 
mined by  taking  moments  about  some  axis 
cutting  the  shaft  center  at  right  angles,  of 
the  components  of  all  forces  in  their  re- 
spective planes  of  revolution,  normal  to  a 
plane  containing  both  this  axis  and  the  shaft 
center  line.  This  is  shown  in  Fig.  219. 

The  center  of  gravity  of  the  upward  forces  is  given  by: 


FIG.  218. 


lu  = 
and  for  the  downward  forcei: 


+ 


The  couple  tending  to  cause  rotation  of  the  shaft  in  an  axial  plane  is 
the  product  of  the  distance  between  the  center  of  gravity  and  the  lesser  of 
the  two  forces,  PI  +  P*  or  P2  -j-  PS.  Let  this  be  denoted  by  P0;  then: 

Q  =  Po(lD  ~  lu)  (7) 


t=5=ii 

r 

if  » 

i      I  1 

h 

j 


FIG.  219. 


the  subscripts  D  and  U  denoting  down  and  up  forces  respectively. 

It  is  often  necessary  to  find  a  resultant  couple  due  to  the  forces  pBx 
and  pBY  of  Fig.  204,  Chap.  XVI;  this  is  given  by: 

Q  =  Vo^Tos  (8) 


340 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


r 


FIG.  220. 


If  the  forces  are  not  in  balance,  there  will  be  a  force  and  a  couple  act- 
ing on  the  engine.  The  force  may  form  a  couple  with  some  other  force 
due  to  the  reaction  of  the  supports,  which  may  act  in  the  same  or  opposite 
direction  of  the  first  couple. 

*  If  all  forces  and  couples  are 

£  not  balanced  in  the  engine  itself 
considered  as  a  free  body,  it  is 
practically  impossible,  due  to 
the  uncertainty  in  locating  the 
origin  of  moments,  to  determine 
the  couple  acting  on  the  sys- 
tem; but  usually  this  is  not 
necessary. 

In  some  engines  it  is  difficult  to  determine  the  maximum  couple  with- 
out plotting  values  for  the  entire  cycle. 

Two-crank  engines  with  cranks  in  the  same  phase  are  treated  the  same 
as  engines  with  a  single  cylinder  so  far  as  inertia  is  concerned.  With 
cranks  at  180  degrees,  the  primary  force  is  balanced,  but  the  small 
imaginary  crank  is  always  at  the  head-end  dead  center  when  the  actual 
crank  is  at  either  center;  consequently  this  gives  an  unbalanced  effect 
equal  to  twice  the  secondary  force.  Referring  to  Fig.  220,  if  PI  is  at  the 
crank-end  dead  center  it  is  smaller  than  P2,  and : 

Q  =  Pi(h  ~  W  W 

If  the  crank  and  pin  are  not  balanced,  their  centrifugal  force,  referred 
to  the  crank-pin  center,  should  be  added  to  PI  in  (9). 

There  is  no  lateral  component  in  the  dead-center  position,  which 
gives  a  maximum  couple.  To  obtain  the  best  results  the  cylinders  should 
be  placed  as  close  together  as  possible,  and  the  cranks  (and  perhaps  a 
portion  of  the  reciprocating  parts)  should  be  balanced. 

Four-crank  engines  such  as  f  3  y_  ? 

Fig.  219,  with  cranks  in  the 

same  plane,  equal  spacing  of      

cylinders  and  weight  of  re- 
ciprocating parts,  are  in  bal- 
ance for  the  primary  force  but 
not  the  secondary  force;  both 
primary  and  secondary  couples  are  balanced.  The  system  as  a  whole 
is  in  no  better  balance  if  the  revolving  parts  are  balanced,  but  bending 
strains  in  frame  and  shaft,  and  pressure  on  bearings  are  relieved  thereby. 

Three-cylinder  engines  with  cranks  opposed  as  in  Fig.  221  have  a 
primary  force  error  unless  the  reciprocating  parts  of  the  center  engine 


FIG.  221. 


BALANCING 


341 


are  equal  to  the  weight  of  the  other  two,  which  is  not  practical.  There 
is  a  secondary  force  error,  but  if  the  cranks  are  evenly  spaced  there  is  no 
couple  error. 

Angle  Engine. — This  is  shown  in  diagram  in  Fig.  222,  with  centers  at 
an  angle  of  90  degrees.  If  the  reciprocating  parts  are  the  same  weight 
in  both  engines,  and  all  the  reciprocating  and  revolving  parts  are  balanced 
by  a  revolving  weight  placed  opposite  the  crank,  the  primary  force  is 
balanced.  There  still  remains  the  unbalanced  secondary  force,  which 
acts  along  a  line  inclined  45  degrees  from  the  line  of  stroke  as  shown  at 
1-2,  Fig.  222,  and  is  a  maximum  as  the  crank  passes  each  dead  center, 
which  is  four  times  a  revolution.  As  both  connecting  rods  engage  with 
the  same  crank  pin  the  unbalanced  axial  couple  is  small.  This  type  of 


FIG.  222. 


FIG.  223. 


engine  is  the  best  balanced  of  any  type  of  2-cylinder  engine  of  usual 
construction. 

Three-crank  engines  with  cranks  at  120  degrees,  cylinders  in  the  same 
plane  and  on  the  same  side  of  the  shaft,  have  no  force  error  due  to  purely 
reciprocating  and  revolving  parts.  It  has  been  shown  that  the  effect 
of  the  reciprocating  parts  along  the  line  of  stroke  is  the  same  as  if  their 
weight  were  attached  to  the  crank  to  produce  the  primary  force.  Then 
assume  three  cranks  at  120  degrees  as  shown  in  Fig.  223  with  such  weights 
attached  Assume  the  cranks  to  move  from  a  symmetrical  position 
through  the  angle  A.  Then  from  trigonometry: 

cos  (A  -f-  B)  =  cos  A  cos  B  —  sin  A  sin  B. 
and: 

cos  (A  —  B)  =  cos  A  cos  B  -\-  sin  A  sin  B. 


342  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Adding  together  gives: 

cos  (A  +  B)  +  cos  (A  -  B)  =  2  cos  A  cos  B. 
lfB  =  60,  2  cos  B  =  1;  and: 

cos  (A  -f  B)  +  cos  (A  -  B)  =  cos  A. 

This  shows  that  the  projection  of  centrifugal  force  upon  the  center 
line  of  engine,  or  any  other  diameter,  of  three  equal  weights  revolving  at 
equal  radii  with  the  same  angular  velocity,  are  in  equilibrium  in  all  posi- 
tions. 

As  the  secondary  force  along  the  line  of  stroke  is  equivalent  to  the 
projection  of  the  centrifugal  force  due  to  the  mass  of  the  reciprocating 
parts  concentrated  at  the  center  of  an  imaginary  crank  pin  traveling  on  a 
circle  R/4L  times  that  of  the  engine  crank,  and  at  a  velocity  twice  as 
great,  it  follows  that  these  imaginary  cranks  are  always  120  degrees  apart 
on  the  crank  circle.  According  to  Fig.  223,  they  are  then  in  equilibrium 
among  themselves.  This  is  true  of  any  revolving  weights,  such  as  the 
cranks,  if  they  are  of  equal  weight  and  their  mass  centers  are  the  same 
distance  from  the  shaft  center. 


As  previously  stated,  the  force  due  to  the  reciprocating  parts  is  not 
caused  by  a  revolving  weight,  and  is  equivalent  to  a  revolving  system 
only  along  the  line  of  stroke.  The  normal  component  FPN,  due  to  inertia 
of  the  reciprocating  parts  is  not  in  balance  in  a  3-cylinder  engine;  this  will 
be  considered  later. 

There  is  a  couple  error,  which  is  a  maximum  for  the  primary  force 
when  the  center  crank  is  at  right  angles  to  the  line  of  stroke.  For  the 
reciprocating  parts  this  is  equal  to  the  product  of  the  inertia,  the  cosine  of 
30  degrees  and  the  distance  between  the  center  of  the  first  and  third 
cylinders.  If  the  cranks  are  not  balanced,  the  radial  inertia  of  crank  and 
pin,  referred  to  the  pin  center,  must  be  added  to  the  inertia.  The  actual 
couple  due  to  reciprocating  parts  is  some  less  due  to  the  secondary  effect. 

Six-crank  engines  with  cylinders  in  the  same  plane  and  on  the  same 
side  of  the  shaft  are  balanced  along  the  line  of  stroke  for  both  primary  and 
secondary  forces,  the  same  as  the  3-crank  engine.  It  is  practically  the 
same  as  two  3-cylinder  engines  placed  end  to  end  as  shown  in  Fig.  224. 

Applying  Formulas  (5),  (6)  and  (7)  shows  that  if  the  arrangement  of 


BALANCING  343 

cylinders  about  the  line  midway  between  the  two  center  cylinders  is 
symmetrical  and  the  reciprocating  parts  and  cranks  weigh  the  same  for 
each  cylinder,  there  is  no  couple  error,  either  primary  or  secondary. 
The  only  advantage  of  balance  weights  in  this  engine  is  to  relieve  frame 
and  shaft  strains  and  bearing  pressures,  and  to  reduce  this  to  a  minimum, 
part  of  the  reciprocating  parts  should  be  balanced.  The  effect  of  the 
forces  acting  on  shaft  and  bearing,  with  and  without  counterbalance  may 
be  seen  in  Figs.  208  and  209,  Chap.  XVI. 

112.  The  Connecting  Rod.  —  Forces  due  to  the  mass  of  the  rod  were 
discussed  in  Chap.  XVI,  and  their  effect  upon  the  frame  may  be  sepa- 
rately considered  if  desired. 

In  computing  counterbalance  it  has  commonly  been  assumed  that  if 
the  rod  is  divided  between  the  reciprocating  and  revolving  parts  inversely 
as  the  mass  center  divides  the  rod,  it  is  correctly  disposed  of.  As  the  pro- 
portion of  reciprocating  parts  to  be  balanced  is  either  fixed  experimentally 
as  in  locomotive  practice,  or  arbitrarily  assumed,  such  a  rule  is  probably 
accurate  enough.  It  is,  in  fact,  correct  along  the  line  of  stroke,  as  may  be 
seen  by  adding 

La    W  I,       LG\W 

-Y-  •  —  a,  toll  -  7^)—  a2 
L       g  \          L  I  g 

where  W  is  the  weight  of  the  rod,  ai  =  Ru2  cos  6,  the  acceleration  of  the 
crank  pin,  and  a2  is  the  acceleration  of  the  crosshead  pin  as  given  by  (30), 
Chap.  XVI,  when  I  is  zero.  The  result  is  identical  with  (30)  and  (32), 
Chap.  XVI  which  gives  the  inertia  of  the  rod  along  the  line  of  stroke 
when  I  =  LQ. 

Normal  to  the  line  of  stroke,  the  force  due  to  the  rod  is  the  algebraic 
sum  of  FYB  and  FXN  given  by  (39)  and  (43),  Chap.  XVI,  the  signs  being 
taken  from  Table  55  of  the  same  chapter.  For  a  rod  of  uniform  section, 
five  cranks  long,  this  gives: 


while  the  divided  rod  method  gives: 

0.5 


g 

an  error  of  42  per  cent. 

Neglecting  the  balance  of  the  crank,  let  it  be  assumed  that  60  per  cent. 
of  the  reciprocating  parts  is  to  be  balanced;  then  the  balance  weight 
required  to  balance  this  according  to  the  usual  method  is: 

W 


344  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

where  WP  is  the  weight  of  the  reciprocating  parts.  If  W  =  kWP,  the 
constant  radial  inertia  (centrifugal  force)  of  the  weight,  assumed  to  have 
its  center  of  gravity  on  the  crank  circle  is  : 


The  inertia  of  the  reciprocating  parts  at  the  head  end  of  the  stroke,  from 
(33),  Chap.  XVI  is: 

-,          «9WPRa* 
r  H  =  1.Z  -  • 
9 

At  the  head  end,  from  (34)  : 


The  inertia  of  the  rod  along  the  line  of  stroke  at  the  head-end  dead  center 
is,  from  (30)  and  (32),  Chap.  XVI: 

,„  =1. 


Q  g 

At  the  crank  end : 


The  total  head-end  inertia  is: 

and  at  the  crank  end: 

(0.8  +  0.9/b)  WpR<** 

Comparing  with  the  counterbalance,  the  amount  unbalanced  at  ends 
of  stroke  is : 

For  head  end,  (0.6  +  0.3fc)  Effli' 

c/ 

For  crank  end,  (0.2  +  0.1  A;) 


g 

Average  for  two  ends,          (0.4  +  0.2k)  - 

J/ 

Normal  to  the  line  of  stroke,  when  the  crank  is  at  right  angles  to  the 
line  of  stroke,  the  inertia  as  just  given,  is: 

7y  =  0.353^=0.353*^.  .".-". 

The  overbalance  in  this  position  is: 

(0.6  +  0.447/b)  WpRu*. 


BALANCING  345 

With  the  divided  rod  method  it  was  assumed  that 


g  9 

This  would  give  an  overbalance  of  : 

(0.6  +  0.3*) 

y 

which  is  less  than  the  actual.     The  ratio  of  the  actual  to  the  assumed  is: 

actual  overbalance       0.6  +  0.447& 
assumed  overbalance       0.6  +  0.3& 

If  k  =  1,  the  ratio  is  1.162.     For  the  Corliss  engine  which  is  being  carried 
through  the  book,  k  =  0.86,  and  the  ratio  is  1.15,  an  error  of  15  per  cent. 

It  may  be  observed  that  the  method  is  more  correct  the  nearer  the 
mass  center  of  the  rod  is  to  the  crank  pin.  This  also  permits  the  rod  to  be 
more  nearly  balanced  by  a  revolving  weight  and  reduces  the  force  and 
couple  errors  due  to  the  rod  mass. 

The  rod  cannot  be  perfectly  balanced  in  engines  of  usual  form,  even 
in  6-crank  engines. 

113.  Turning  Effort.  —  The  turning  force  is  transmitted  to  the  engine 
frame  at  the  guide  (or  cylinder,  when  a  trunk  piston  is  used),  giving  the 
force  pA.  This  is  plotted  for  the  2*0-  by  48-  in.  Corliss  engine  in  Fig.  206, 
Chap.  XVI.  It  forms  a  couple  equal  to  the  turning  couple,  but  may  be 
offset,  at  least  for  part  of  the  cycle,  by  the  effect  of  counterbalance  and 
other  forces  normal  to  the  line  of  stroke  acting  at  the  shaft  center.  This 
may  be  seen  in  Fig.  206,  Chap.  XVI,  which  is  drawn  to  the  same  force  scale 
as  Fig.  205.  These  forces  are  periodic  and  tend  to  cause  vibration 
normal  to  a  plane  containing  the  engine  and  shaft  center  lines  ;  they  may 
not  well  be  balanced. 

In  many  engines  with  massive  frames  and  heavy  foundations  the 
effect  of  turning  effort  is  unimportant  and  is  seldom  given  any  thought. 
In  ships  and  automobiles,  in  which  the  engines  are  made  as  light  as  prac- 
ticable, vibration  may  be  considerable,  due  to  this  cause,  but  is  offset  in 
part  by  arranging  cylinders  and  cranks  so  as  to  obtain  as  uniform  a 
turning  effort  as  possible;  that  is,  to  provide  more  impulses  per  cycle. 
The  vibrating  forces  are  then  more  frequent  but  vary  less  from  the  mean 
force  applied. 

With  uniform  turning  effort  as  given  by  steam  turbines  and  electric 
motors,  the  couple  is  constant,  and  this  condition  is  approximated  in 
multi-cylinder  engines. 

As  previously  stated,  the  nature  of  supports  or  foundation  determines 
to  a  great  extent  the  effect  of  unbalanced  forces.  If  the  periodic  motion  is 
in  synchronism  with  the  natural  period  of  vibration  of  the  supports  the 


346  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

consequences  may  be  serious;  but  if  not,  a  small  amount  of  unbalanced 
force  will  usually  have  no  serious  effect. 

114.  Balance  weights  are  of  various  forms,  some  of  which  are  shown 
in  Chaps.  XXVII  and  XXVIII.  The  required  balance  referred  to  the 
crank  pin  may  be  determined,  then  when  the  general  form  of  the  counter- 
weight is  selected,  it  may  be  drawn  to  scale  with  assumed  dimensions. 
With  certain  simple  forms  the  center  of  gravity  may  be  calculated ;  then 
the  area  may  be  calculated  or  found  by  a  planimeter  and  the  moment 
determined. 

Some  designers  prefer  to  draw  the  weight  to  scale,  then  cut  it  out  of 
card  board  or  thin  wood  and  either  balance  it  on  a  point,  or  hang  it  from 
two  different  points  so  that  it  is  free,  allowing  a  plumb  line  to  hang  from 
the  point  of  suspension;  the  point  where  the  lines  cross  in  the  two  posi- 
tions is  the  center  of  gravity. 

Sometimes  the  first  trial  will  suffice,  any  deviation  from  the  required 
weight  being  provided  for  by  varying  the  thickness. 

When  there  is  not  room  to  accommodate  the  required  weight,  the 
casting  is  sometimes  cored  and  filled  with  lead. 

The  segment  of  a  circular  ring  is  a  form  rather  commonly  used,  and 
this  lends  itself  to  simple  means  of  finding  dimensions.  Fig.  225  shows  a 
crank  with  a  counterweight  of  this  kind.  The  moment  of  the  counter- 
weight about  the  shaft  center  must  equal  the  sum  of  the  moments  of  all 
weights  to  be  balanced ;  or : 

wABtlB  =  S(TFZ).  =  M  (10) 

where  w  is  unit  weight,  AB  the  area  of  the  segment  forming  the  counter- 
balance, t  its  thickness  and  1B  the  distance  of  its  center  of  gravity  from  the 
shaft  center.     All  dimensions  must  be  either  in  feet  or  inches;  if  the  latter, 
w  is  the  weight  per  cu.  in. 
From  (10): 


Without  taking  space  for  the  derivation  of  the  formula: 

•  sin  »• 


r0  +  r 
It  is  plain  that: 


Then  letting  r0/V  =  q: 


AB1B  =  ?(r03  -r3)sin|  =  |r3  (g3  -  1)  sin  |  (12) 


BALANCING 
For  a  plain  balance  with  no  limits  for  r0, 


347 


=     3 


1  + 


3AB1B 
2r3  sin 


1  + 


(13) 


FIG.  225. 

Then: 

r0  =  qr  (14) 

If  a  disc  crank  is  used  which  fixes  both  r  and  7*0,  and  if  t  is  assumed: 


sin  ^  = 


If  6  is  assumed: 


-  1) 


t  = 


3M 


-  1    sn 


(15) 


(16) 


With  a  disc  crank,  the  thickness  of  the  disc  should  not  be  counted  as 
part  of  either  arm  or  balance  thickness. 


348 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


In  the  Corliss  engine  for  which  data  is  tabulated  in  Chap.  XVI,  the 
values  of  pR  from  Table  57  were  taken  at  crank  positions  6,  12,  18  and 
24,  and  averaged.  This  method  is  arbitrary,  and  in  this  case  gave 
slightly  smaller  weight  than  by  taking  0.6  of  the  reciprocating  parts  added 
to  the  revolving  parts,  the  rod  being  divided  inversely  as  the  mass  center. 
The  average  gives  the  radial  inertia  required  of  the  counterweight  and  is  : 

10,366  +  20,580  +  4126  +  25,385 
— 


From  (1),  solving  for  W  (WB  in  this  case)  gives  the  weight  referred  to 
the  crank  pin;  or: 

_gCP_  32.16X15,114     _ 
R-         2  X  109 

Then  from  (10),  in  pound-inches. 

M  =  2225  X  24  =  53,400 

From  (13),  more  conveniently  taking  dimensions  in  inches,  and  assum- 
ing a  cast-iron  crank: 


Then: 


a  =  si  1    ,     3  X  53,400 

\       r  2  X  0.26  X  6.375  X  123  X  0.866 


r0  =  3.22  X  12  =  38.75  in. 


This  assumes  the  thickness  of  balance  as 
6^6  in.,  which  is  the  same  as  the  arm 
thickness  for  the  crank  designed  for  the 
20-in.  Corliss  engine  in  Chap.  XXVII. 
The  angle  d  was  taken  as  120  degrees. 

The  crescent  and  the  segment  of  a  circle 
are  forms  used  in  locomotive  balancing. 
Fig.  226  shows  a  crescent,  the  distance  be- 
tween the  centers  of  the  two  arcs  being 
denoted  by  lc.  AB  is  the  area  of  the 
crescent  and  A  the  area  of  the  segment 
between  this  and  the  chord.  A  simple 

exact   formula  for  1B  was  published  by  the  author  in  the  American 

Machinist  some  years  ago  and  is : 

A  7 

(17) 


FIG.  226. 


or: 


Alc  =  AB1B  =  — 


(18) 


BALANCING  349 

If  the  counterbalance  is  a  segment,  A  =  0  and  lc  =  °°  ;  then  the  formula 
fails;  then: 


or: 

A.JL,  -  £  (20) 


CHAPTER  XVIII 
REGULATION  DURING  THE  CYCLE.     FLYWHEELS 

115.  Introduction. — It  is  evident  from  the  turning-effort  diagrams  of 
Par.  106,  Chap.  XVI  that  the  force  applied  to  the  crank  pin  in  the  direction 
of  motion  varies  considerably  during  the  engine  cycle,  even  when  a  number 
of  cylinders  are  employed.  Unless  the  resistance  varies  in  identically 
the  same  way,  which  it  never  does,  a  change  of  velocity  must  occur  during 
the  cycle,  the  number  of  fluctuations  depending  upon  the  number  of 
times  the  turning  effort  varies  from  the  mean;  also  the  displacement  of 
the  crank  pin  from  the  place  it  would  occupy  at  any  instant  with  per- 
fectly uniform  motion,  occurs. 

The  function  of  the  flywheel  is  to  limit  speed  fluctuation  and  dis- 
placement to  an  amount  found  by  practice  to  give  satisfactory  operating 
results. 

The  method  of  controlling  speed  fluctuation  is  simpler  and  is  much 
more  commonly  used  for  designing  wheels  for  all  purposes.  The  dis- 
placement method  is  more  difficult  and  is  used  only  when  engines  are 
designed  to  drive  alternating-current  generators  which  are  to  operate  in 
parallel.  The  relation  between  the  two  methods  will  be  shown  in  Par. 
118. 

In  applying  either  of  these  methods  to  practical  work  certain  assump- 
tions are  made  and  refinements  neglected,  as  the  varying  condition  of 
operation  of  any  type  of  engine  does  not  permit  of  great  accuracy.  At- 
tention is  called  to  these  approximations  in  the  proper  places. 

Notation. 

D  =  diameter  of  wheel  in  feet. 
R  =  radius  of  crank  circle  in  feet. 
r  =  radius  of  gyration  in  general,  in  feet. 
w  =  weight  in  pounds  in  general. 
WR  =  weight  of  wheel  rim  in  pounds. 
Ww  =  total  weight  of  wheel  in  pounds. 

W  =  weight  in  pounds  concentrated  at  crank-pin  center,  which  would 
give  the  same  effect  as  the  weight  of  the  entire  wheel  concen- 
trated at  its  center  of  gyration. 

350 


FLY  WHEELS  351 

M  =  mass  assumed  concentrated  at  center  of  crank  pin,  corresponding 

to  W  (  =  W/g). 

F  =  turning  effort  in  pounds,  above  or  below  the  mean  effort. 
E  =  mean  energy  per  piston  stroke  in  foot  pounds. 
AE  =  fluctuation  of  energy  above  or  below  the  mean;  usually  the 

maximum  =  eE. 
H  =  indicated  horsepower. 
V  =  mean  velocity  of  rim  in  feet  per  second. 
Vi  =  maximum  velocity  of  rim. 
Vz  =  minimum  velocity  of  rim. 
v  —  velocity  of  crank  pin  in  feet  per  second. 
N  =  r.p.m. 

a  =  linear  acceleration  of  crank  pin  in  feet  per  second  per  second. 
t  =  time  of  one  engine  cycle  in  seconds. 

s  =  displacement  in  feet  of  the  crank  pin  from  its  mean  position, 
c  =  coefficient  of  fluctuation  of  energy. 
d  =  coefficient  of  fluctuation  of  velocity. 
A  =  change  of. 

a  =  displacement  of  crank  pin  from  mean  position  in  electrical  degrees. 
c  =  number  of  cycles  of  alternator  per  second. 
p  =  number  of  poles. 

m  =  time  scale  =  number  of  seconds  per  inch. 
n  =  force  scale  =  number  of  pounds  per  inch. 
q  =  scale  of  Mv  =  number  of  units  of  Mv  per  inch. 
k  =  scale  of  Ms  —  number  of  units  on  Ms  per  inch. 
x,  y  and  I  =  measurements  on  diagrams  in  inches. 
C  and  K  =  constants  in  Formula  (8)  and  Table  60. 
g  =  32.16. 

116.  The  Control  of  Speed  Fluctuation.— This  method  may  be  applied 
where  the  resistance  varies,  so  long  as  the  cycle  is  constant  at  a  given 
load.  In  most  cases  where  such  calculations  are  made  the  resistance  is 
assumed  to  be  uniform  for  a  given  load,  so  that  the  turning  effort  dia- 
gram shown  in  Fig.  194,  Chap.  XVI  will  be  used  to  illustrate  the  methods 
of  determining  flywheel  weight  in  this  chapter.  This  is  reproduced  in 
Fig.  227. 

As  it  is  the  variation  from  mean  which  causes  change  of  speed,  the 
areas  above  and  below  the  mean  effort  line  are  the  areas  considered,  and 
these  are  shaded.  Each  one  of  these  lobes  bounded  by  the  curve  and 
the  mean  line  represents  the  fluctuation  of  energy  as  the  crank  pin  passes 
between  two  intersections  of  these  lines,  and  is  denoted  by  AJ£.  The 


352 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


crank  pin  has  a  certain  velocity  at  A  (and  the  flywheel  rim  a  proportion- 
ate velocity).  In  passing  from  A  to  B  energy  is  taken  from  the  wheel 
equivalent  to  the  area  of  the  lobe  between  A  and  B,  and  its  velocity  is 
reduced.  At  B  the  velocity  begins  to  increase  and  continues  to  do  so  until 
C  is  reached,  the  kinetic  energy  stored  in  the  wheel  during  this  time,  as  the 
crank  pin  travels  from  B  to  C  being  represented  by  the  lobe  between  B 
and  C. 

Kinetic  energy  varies  as  the  square  of  the  velocity;  then  it  is  obvious 
that  the  maximum  fluctuation  of  velocity  is  caused  by  the  maximum 
fluctuation  of  energy,  and  the  largest  lobe  AJ57  must  be  used  in  the  follow- 
ing calculations. 

Let  E  be  the  energy  of  the  cycle  in  ft.  Ib.  divided  by  the  number  of 
strokes,  or  the  mean  energy  per  stroke.  Then  the  coefficient  of  the  fluc- 
tuation of  energy  is : 


e  — 


E 


(1) 


FIG.  227. 

Basing  c  upon  the  energy  per  stroke  has  advantages  over  energy  per  cycle 
in  simplifying  the  final  equations." 

If  N  =  r.p.m.,  there  are  2N  strokes  per  minute;  the  i.h.p.  then  is: 

2NE          NE 
~  33,000  ~  16,500* 
Then  for  all  engines  the  mean  energy  per  stroke  is : 


N 

As  just  stated,  the  change  of  kinetic  energy  of  the  rim  due  to  the 
maximum  fluctuation  of  velocity  from  V%  to  V\  ft.  per  sec.,  or  vice  versa, 
is  equivalent  to  the  maximum  fluctuation  of  energy  along  the  crank- 
pin  path;  then  from  (1): 

(Vl2  _  y22)       WR(Vi  -  F2)(Fi  +  F2) 


&E  =  eE  = 


(3) 


2g  2g 

If  V  is  the  mean  velocity  and  8  the  coefficient  of  the  fluctuation  of  speed: 

(4) 


FLY  WHEELS 
Also,  nearly  enough  for  practical  purposes: 


353 


or: 


2  60 

=  7rDN 

30 
where  D  is  the  wheel  diameter  in  ft. 

Substituting  (2),  (4)  and  (5)  in  (3)  gives: 
16,500e# 


(5) 


N 


Solving  for  WR  gives : 


60  X  30  X  20 


WR  =  194,000,000 


(6) 


Stresses  in  flywheels  will  be  treated  in  Chap.  XXX,  but  to  limit  these  to 
safe  values  the  rim  speed  is  not  allowed  to  exceed  a  certain  limit;  a  com- 
mon speed  allowed  for  cast-iron  wheels  is  one  mile  (5280  ft.)  per  minute. 
Then: 

irDN  =  5280  and  DN  =  1680. 

Substituting  this  in  (6)  gives : 


Formula  (7)  is  convenient  in  estimating  when  it  is  likely  that  the 
wheel  will  run  up  to  the  limiting  speed;  but  it  gives  a  lighter  rim  than 
does  (6)  should  the  rim  velocity  be  reduced,  and  this  must  be  kept  in 
mind. 

If  values  of  e  and  5  are  known,  (6)  and  (7)  may  be  written: 

W    -  C    H      -  K—  rtrt 

-  °  D2N3  ~    *•  N 

The  value  of  e  in  Fig.  227  is  0.33.  This  was  plotted  for  the  20-  by  48- 
in.  Corliss  engine  designed  in  Chap.  XII  with  a  J^  cut-off.  Goodman 
gives  the  values  of  e  for  double-acting  steam  engines  in  Table  58. 

TABLE  58 


Cut-off 

Single  cylinder 

Two-  cylinder  cranks  at  90 

Three-  cylinder    cranks    at 
120 

0.1 

0.35 

0.088 

0.040 

0.2 

0.33 

0.082 

0.037 

04 

0.31 

0.078 

0.034 

0.6 

0.29 

0.072 

0.032 

0.8 

0.28 

0.070 

0.031 

Full 

0.27 

0.068 

0.030 

354 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


It  will  be  seen  that  e  decreases  when  the  cut-off,  and  therefore  the  load 
increases;  but  not  in  the  same  ratio.  It  would  seem  from  this  that  a 
value  of  H  near  the  maximum  should  be  used.  It  is  &E  (which  varies 

as  eH)  that  causes  the  fluctuation  of  speed,  and  not  e  (  =  -g-j  • 

For  4-cycle  internal-combustion  engines  working  on  the  Otto  cycle, 
Goodman  gives  the  values  in  Table  59. 

•      TABLE  59 


Exploding  at 

Values  of  e  for  internal-combustion  engines 

Single-acting 

Double-acting 

1-cyl. 

2-cyl. 

4-cyl. 

1-cyl. 

2-cyl. 

Every  cycle  
Alternate  cycles  .  .  . 

3.7to4.5 
8.5  to  9.8 

1  .  5  to  1  .  8 
2  .  5  to  3  .  0 

0.3  to  0.4 

2.3  to  2.8 

0.3  to  0.4 

TABLE  60 


Service 

e 

3 

C 

K 

Pumping  .  . 

0.4 

Ho 

2,320,000,000 

824 

Machine  shop  
Textile  paper  and  flour  mills 

0.4 
0  4 

Ks 

Ho 

2,710,000,000 
3  100  000  000 

960 
1  100 

Spinning  machinery  

0.4 

Moo 

7,740,000,000 

2,750 

Gas  compression        

0.4 

Moo 

7,740,000,000 

2750 

Rolling  mill 

2  0 

Ho 

11620000  000 

4  110 

Cable  railway  

0.4 

M20 

9,300,000,000 

3,300 

Electrical  machinery  (belted) 

0  4 

Mso 

11,620  000  000 

4  110 

Direct-connected  alternator 

0  4 

•  y$v 

387  500  OOOp 

140p 

Sugar  mill  —  grinder  
Sugar  mill  —  crusher  

0.4 
0.4 

Ha 

>73 

3,200,000,000 
5,600,000,000 

r 

1,160 
2,010 

In  Chap.  XIV,  Par.  78,  reference  was  made  to  hit-and-miss  governing  at 
light  loads.  From  (6)  and  (7)  it  may  be  seen  that  the  speed  fluctuation 
of  a  wheel  of  given  weight  varies  as  the  product  eH.  If  every  alternate 
impulse  is  missed  the  value  of  H  is  one-half  full  load.  Then  taking  values 
from  Table  59,  to  give  the  same  regulation  at  half  load  a  single-cylinder 
gas  engine  would  need  a  wheel  10  per  cent,  heavier,  and  a  two-cylinder 
engine  83  per  cent,  as  heavy  as  for  full  load.  This  may  not  hold  true  for 
lighter  loads,  but  by  making  a  wheel  on  a  hit-and-miss  engine  some  20 
per  cent,  heavier  than  the  formula  indicates,  fair  regulation  should  be 

23 


FLY  WHEELS  355 

obtained  for  quite  a  wide  range  of  load.  If  combination  governing  were 
used,  the  hit-and-miss  principle  would  not  be  applied  except  at  the  lower 
range;  if  at  one-half  load,  one-quarter  load  could  be  carried  by  missing 
every  alternate  impulse. 

Many  tables  of  6  have  been  published  and  there  is  considerable 
variation.  Table  60,  for  double-acting,  single-cylinder  steam  engines  is 
given  as  an  example  in  tabulating  C  and  K  of  Formula  (8),  also  to  give 
some  values  of  d  which  have  been  taken  from  various  sources.  The  value 
for  the  direct-connected  alternator  will  be  further  explained  in  Par.  3 ;  p 
denotes  the  number  of  poles.  The  table  has  been  used  by  the  author  for 
several  years  and  part  of  the  data  are  from  his  practice.  It  must  be 
used  with  judgment;  in  some  cases  the  values  of  C  and  K  may  give  too 
small  values  to  enable  the  proper  construction  of  a  wheel  for  strength, 
especially  if  the  rim  speed  is  near  the  maximum  limit. 

The  value  of  c  is  taken  a  little  high  to  provide  for  discrepancies. 
From  Table  58  it  will  be  seen  that  the  value  of  e  varies  inversely  as  the 
square  of  the  number  of  alternations  of  turning  effort.  This  would  mean 
that  a  cross-compound  steam  engine  requires  a  wheel  but  one-quarter  as 
heavy  as  a  single-cylinder  engine  of  the  same  power.  But  with  compound 
engines,  load  distribution  between  high-  and  low-pressure  sides  is  con- 
siderably affected  by  different  valve  settings,  and  this  in  turn  affects  the 
form  of  the  turning  effort  diagram,  often  increasing  the  value  of  AE. 
Then  too,  a  compound  engine  does  not  respond  to  the  governor  quite  as 
quickly  as  a  simple  engine,  especially  if  the  high-pressure  cylinder  only 
is  directly  governed.  It  therefore  seems  that  some  allowance  should  be 
made  in  using  values  of  e  found  from  carefully  designed  compound  dia- 
grams. With  large  units  which  are  to  have  frequent  applications  of  the 
indicator  this  need  not  be  very  much,  possibly  an  increase  of  25  per  cent. 
In  a  good  many  cases  the  author  has  determined  wheel  weights  by  the 
use  of  Table  60  for  a  simple  engine,  and  taken  from  75  to  100  per  cent,  of 
this  weight  for  cross-compound  engines  of  the  same  power,  and  while 
this  may  seem  excessive,  it  has  been  satisfactory. 

For  internal-combustion  engines  with  more  than  one  cylinder  it  is 
probably  not  necessary  to  make  much  allowance,  although  there  may  be 
a  small  difference  in  the  indicator  diagrams  of  the  different  cylinders; 
increasing  the  value  of  e  about  10  per  cent,  over  the  value  found  with 
equal  indicator  diagrams  will  probably  give  ample  allowance. 

For  strict  accuracy  the  diameter  D  used  in  the  formulas  should  be 
equal  to  twice  the  radius  of  gyration  of  the  entire  wheel,  or,  more  strictly, 
of  all  parts  revolving  on  the  shaft,  and  WR  should  include  the  entire 
weight  of  wheel  and  other  revolving  parts.  Jf  w  is  the  weight  of  a  part 


356  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

of  the  wheel  or  other  revolving  part  and  r  its  radius  of  gyration  about  the 
shaft  center,  then: 

Wa  (f)2    :    S(w2)  (9) 

The  calculation  of  wr2  requires  much  time  and  is  usually  an  unnec- 
essary refinement.  If  D  is  taken  as  the  outside  diameter  of  the  wheel, 
the  weight  of  the  wheel  rim  found  by  (6),  (7)  or  (8),  and  the  weight  of 
arms,  etc.  neglected  in  the  calculation,  the  energy  of  the  wheel  will  be 
equal  to  or  greater  than  that  required  if  it  is  a  belt  wheel;  for  a  heavy 
flywheel  with  a  thick  rim  (radially)  the  weight  may  fall  a  little  short  if 
calculated  in  this  way,  but  increasing  the  weight  so  found  by  10  per  cent, 
will  make  ample  provision.  The  value  of  e  in  Table  60  for  simple  steam 
engines  will  cover  this. 

In  estimating  it  is  convenient  to  know  the  weight  of  the  entire  wheel. 
After  finding  rim  weight  WR  from  (6),  (7)  or  (8),  sufficient  metal  will  be 
allowed  for  a  well-built  wheel  of  the  more  common  designs  if  the  total 
weight  W  is  as  follows : 

For  belt— or  rope  wheels.  ....   FV  =  1.7  to  1.8TF«         (10) 
For  heavy-rimmed  flywheels     .    .    Ww  =  1.5tol.6WR         (11) 

117.  The  Control  of  Displacement. — Alternators  operating  in  par- 
allel must  be  in  synchronism  to  give  good  results.  They  must  not  only 
have  the  same  r.p.m.  but  the  angular  distance  between  corresponding 
poles  (in  electrical  degrees)  must  not  exceed  a  certain  limit  or  cross 
currents  will  be  set  up,  lowering  the  average  voltage  and  causing  loss  of 
effectiveness  and  efficiency. 

The  general  effect  of  irregularity  of  turning  effort  upon  speed  fluctua- 
tion during  the  cycle  has  been  discussed  in  the  preceding  paragraph; 
it  now  remains  to  study  the  effect  upon  the  displacement  of  the  wheel 
from  some  mean  position  it  would  occupy  if  revolving  with  a  perfectly 
uniform  velocity. 

To  try  and  make  this  clear  before  proceeding  with  a  mathematical 
determination  of  the  displacement  curve,  a  rather  imaginary  piece  of 
apparatus  shown  in  Fig.  228  will  be  examined.  Assume  two  wheels 
running  side  by  side  with  the  same  r.p.m.  Wheel  No.  1  runs  with 
absolute  uniformity  of  speed  and  has  on  it  two  paper  drums  operated 
by  gears  with  a  uniform  motion.  The  arrow  shows  the  direction  the 
paper  moves.  Wheel  No.  2  is  an  engine  flywheel  subject  to  the  variation 
of  turning  effort.  Arm  A,  containing  a  pencil  is  fastened  to  a  spoke  of 
this  wheel.  During  the  revolution  the  relative  angular  position  of  the 
two  wheels  changes,  causing  the  pencil  to  move  across  the  spoke  of  wheel 


FLY  WHEELS 


357 


1  and  draw  a  curve  as  the  paper  moves  at  right  angles  to  the  pencil  move- 
ment. This  is  a  displacement  curve  and  with  a  constant  load  on  the 
engine  it  should  be  exactly  alike  for  each  cycle  of  the  engine. 

A  line  drawn  in  the  direction  of  paper  motion  in  such  a  way  that  the 
algebraic  sum  of  the  areas  enclosed  between  it  and  the  curve  for  a  com- 
plete cycle  is  zero,  is  the  mean  line;  this  is  shown  dotted,  and  if  the  pencil 
of  wheel  2  is  brought  over  to  this  line  it  will  show  the  mean  position  of 
wheel  2  relative  to  wheel  1  as  the  two  wheels  revolve. 


FIG.  228. 

A  similar  curve  may  be  drawn  from  the  crank-effort  diagram  of  any 
engine.  When  the  turning  effort  is  above  or  below  the  mean  it  causes 
positive  or  negative  acceleration  of  the  wheel.  It  is  convenient  to  refer 
all  force  and  motion  to  the  crank-pin  center.  Then  if  M  is  the  mass 
( =  W/g,  where  W  is  the  weight  in  Ib.)  giving  the  same  effect  as  the  wheel 
concentrated  on  the  crank  circle,  a  the  linear  acceleration  of  the  crank 
pin  and  F  the  turning  effort  in  Ib.  above  or  below  the  mean : 

F  =  Ma 


or: 


a=M 


(12) 


Then  a  is  directly  proportional  to  F,  and  Fig.  227  may  be  converted 
into  an  acceleration  diagram  by  changing  the  scale. 


358 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


In  practice  the  displacement  is  very  slight,  so  that  with  the  usual 
spacing  of  the  crank  circle,  if  it  be  assumed  that  equal  increments  of  time 
are  accompanied  by  equal  increments  of  space  as  the  crank  moves  around 
the  circle,  the  degree  of  accuracy  will  be  in  keeping  with  that  of  the  whole 
method.  Then  by  changing  the  scale,  space  may  be  changed  to  time; 
then  the  time  of  one  cycle  in  seconds  which  is  represented  by  the  length 
of  the  diagram  is: 

For  a  2-stroke  cycle,  t 


For  a  4-stroke  cycle,  t 


60 

N 
120 

N 


(13) 


Fig.  227  is  reproduced  in  Fig.  229  as  an  acceleration-time  diagram. 
The  dimensions  x,  y  and  I  are  always  in  inches,  and  areas  in  sq.  in. 


MEAN  LINE 


PLOT  LINE 


\0    I      2     3     4     S    6 

FIG.  229. — Acceleration-time  diagram. 

A  general  expression  for  acceleration  is: 

dv 
dt 


a  =  -T. 


or: 


dv  =  a-dt 


(14) 


It  is  then  obvious  that  by  integrating  the  acceleration-time  curve  above 
and  below  the  mean  line,  a  curve  may  be  plotted  with  change  of  velocity 
as  ordinates  and  time  as  abscissas — a  velocity-time  diagram. 
A  general  expression  for  velocity  is: 

ds 

dt 


V  =  -r. 


or: 


ds  =  v  •  dt 


(15) 


After  finding  the  mean  line  for  the  velocity-time  curve,  this  curve  may  be 
integrated  and  a  displacement-time  curve  plotted.  From  the  mean  line 
found  for  this  curve  the  maximum  displacement  from  mean  may  be  found. 


FLY  WHEELS 


359 


All  integrations  will  be  made  with  a  planimeter,  the  mathematical  ex- 
pressions being  used  to  make  the  operations  clear. 

Fig.  230  is  a  velocity-time  curve  and  Fig.  231  a  displacement-time 
.'iirve. 

To  make  practical  use  of  these  diagrams  scales  must  be  used  for  the 
different  quantities,  and  it  avoids  much  confusion  if  these  are  determined 
a^  the  work  proceeds. 


MEAN  LINE 


PLOT  LINE 


FIG.  230. — Velocity-time  diagram. 

For  all  curves:  1  in.  =  m  seconds  =  j,  where  t  is  found  from  (13). 


Then: 

dt  =  mdx 
where  dx  is  in  inches. 

Let:  1  in.  =  n  Ib.  =  n  units  of  F  (=  Ma). 

Or:  1  in.  =  -      units  of  a  =  -     ft.  per  sec.2 


(16) 


FIG.  231. — Displacement-time  diagram. 

n  has  already  been  determined  for  plotting  Fig.  227.     Then: 


(17) 


To  integrate  the  acceleration-time  curve  to  obtain  change  of  velocity, 
the  mathematical  expression  may  be  reduced  by  substituting  (16)  and 
(17)  in  (14)  as  follows: 

w  rv-u*  (18) 


360  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  quantities  n  and  m  are  known.  If  it  is  desired  to  find  the  dis- 
placement of  a  wheel  already  designed  or  built,  M  is  also  known.  The 
quantity  in  the  integration  sign  is  the  area  in  sq.  in.  between  any  two 
ordinates  found  by  the  planimeter.  If  this  is  the  first  space  of  Fig.  229, 
shown  shaded,  it  is  below  the  mean  line  and  must  be  laid  off  at  point  1  of 
Fig.  230  from  and  below  the  plotting  line,  the  mean  line  not  having  been 
determined.  The  mean  line  is  found  by  taking  the  difference  of  the  areas 
above  and  below  the  mean  line  and  dividing  by  the  length  in  inches. 
If  the  area  above  the  line  is  greater,  the  mean  line  will  be  drawn  above 
the  plotting  line,  or  vice  versa.  If  y\  is  the  distance  from  mean  to 
plotting  line  and  3/2  from  plotting  line  to  curve,  this  may  be  expressed 
algebraically  thus: 

y\  =   I  y^'dx  +  I  (19) 

To  fix  the  scale  for  the  velocity-time  curve,  determine  the  total  height 
that  it  is  desired  to  make  it,  then  add  together  the  maximum  and  mini- 
mum values  of  Az;  (not  algebraically)  and  divide  by  the  desired  height  in 
inches  and  round  up  to  some  convenient  figure. 

In  most  design  problems  M  is  unknown  and  may  be  carried  through 
the  calculation  as  an  unknown  quantity.  We  may  then  write  (18)  as 
follows: 

fxt 

M-kv  =  nm  I     y  dx  (20) 

Jxi 

Then  let:  1  in.  =  q  units  of  Mv; 


1  in.  =  -^units  of  v  =  ~  ft.  per  sec. 


Then: 


To  integrate  the  velocity-time  curve  to  obtain  displacement  the  ex- 
pression is  found  by  substituting  (16)  and  (21)  in  (15),  thus: 

f*tt  f*X2  m      /»X2 

As  =       vdt  =         Ifyn-dx  =  5?       ydx  (22) 

Jti  Jxi    ™  MJxi 

If  M  is  unknown: 

f* 

M-As  =  qm  I    ydx  (23) 

Jxi 

The  scale  is  found  as  for  the  velocity-time  curve  ;  then  let  : 
1  in.  =  k  units  of  Ms; 


FLY  WHEELS  361 

or: 

k        •*      t 
1  in.  =  -=-=  units  of  5. 

M 

The  mean  line  is  found  as  before  from  (19). 

The  maximum  ordinate  yM  from  the  mean  line  must  now  be  found 
and  this  corresponds  to  the  greatest  displacement  SM  in  feet.  Then: 

•"'Jf-V  ;  (24) 

If  M  is  unknown  and  to  be  determined  : 

M-^i  ••••!•  .          (25) 

SM 

The  value  of  SM  is  determined  from  electrical  conditions.  There  are 
360  electrical  degrees  between  two  poles  of  the  same  sign.  If  the  alter- 
nator has  p  poles,  the  number  of  electrical  degrees  in  the  entire  circle  is  : 

360-|  =  180p. 

If  the  allowable  displacement  either  side  of  mean  is  a  electrical  degrees 
(degrees  of  phase),  the  maximum  allowable  displacement  of  the  crank 
pin  in  feet  is: 

' 


There  seems  to  be  a  scarcity  of  data  regarding  a  in  electrical  engineer- 
ing hand  books,  but  2.5  is  given  in  papers  from  the  Trans.  A.S.M.E. 
referred  to  at  the  end  of  this  chapter. 

•     If  the  number  of  electrical  cycles  per  second  is  denoted  by  c,  the  rela- 
tion between  the  cycle,  number  of  poles,  and  speed  is  given  by: 

120c 


It  is  convenient  to  tabulate  data  as  the  work  proceeds  as  shown  in 
Table  61.  The  minus  sign  indicates  that  the  larger  part  of  the  area  up  to 
that  number  is  below  the  mean  line;  when  it  is  above  the  mean  line  the 
sign  is  plus.  Obviously  the  final  value  should  be  zero,  bringing  the  curve 
back  again  to  the  plotting  line  to  the  point  from  which  it  started.  In 
determining  the  areas  for  the  table,  the  planimeter  may  be  started  at  the 
beginning  of  the  curve  each  time  if  the  diagram  is  not  too  large;  or,  each 
division  may  be  measured  and  added  algebraically. 

From  (6),  the  weight  varies  inversely  as  the  square  of  the  diameter 


362 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


<N 

<N    O 

co  co 

d  *" 

>O    00 
<N    iO 

,_, 

8§ 

8°. 

* 

<N   "5 

0 

58 

TH    Th 

go 

<N   O 

05 

TH 

O   <M 

00 

<N   O 

O^    CO 

O5   "3 

TH      |> 

T-H 

o  ^ 

- 

.£.1 

Tfl      IO 

«b 

d  ^ 
1     1 

co  »o 

CO    CO 

1   1 

« 

TH 

*8 
1   1 

«0   0 

CO   00 

Jg° 

1 

7  7 

ec 

83 

d  ^ 

1 

iO   t> 

•    CO 

TH     CO 

1   1 

00    /— 

™  § 

co  10 
oo     • 

<N 

d  w 
1     I 

0   <N 

1   1 

8-0 

CO    10 

_< 

d  <N 

0    00 

1     1 

1   1 

» 

"S 

a 

0 

s 

S  ! 

li 

2     °° 

• 

O  O  O  O 

n 

CO            CO    ^D 

^  ^  o  10 

d  ^  d  TH 

£-S£ 

8 

d  °°  o  d 

CO    iO   CO    iO 

^ 

o       o  co 

1     1 

o 

IN 

d^o  ® 

1   1 

CO   !>•    C^l    iO 

0          00 

1   1 

00 

co^oS^ 

1-1 

•    rH    O    ^f 

0       1    1 

CO 

o  »o 

s 

0  °  0  TH' 

1       | 

CO           00   O 
"*   1C  <N   O 

0   <N   0   l> 

1   1 

2 

CO  O  O5  C^ 

d  co     •  t^ 

1  1  °~ 

co  J2  os  oo 

" 

d  k  **  ^ 

|         ,      TH    CO 

^1    (^ 

CO   l>   T^   iO 

CO 

o          •  oo 

1            1         TH      TJH 

1         1 

£ 

03 
fl 

o 

03      :     03 

FLY  WHEELS  363 

at  which  it  is  applied ;  then  if  the  weight  ( =  gM )  assumed  concentrated  at 
the  radius  R  of  the  crank  circle  is  referred  to  diameter  D,  we  have : 


or: 

W«=w(^y  (28) 

The  general  notes  in  reference  to  weight  of  arms,  etc.  of  the  preceding 
paragraph  apply  to  wheels  designed  by  the  displacement  method  also. 
It  would  be  a  farce,  however,  to  take  the  trouble  to  calculate  a  wheel  by 
this  method  for  a  simple  engine,  to  use  on  a  cross-compound  engine. 

118.  Comparison  of  Methods. — For  a  given  engine  with  certain  condi- 
tions of  speed,  steam  pressure,  valve  setting,  etc.,  Formula  (6)  may  be 
written : 

WR=  constant  (2g) 


Also  from  (25),  (26)  and  (28): 


WR=  g~  (30) 

constant 


Equating  (29)  and  (30)  we  may  write: 

pd  =  constant 
or: 

1 


p  X  constant 

This  is  the  form  for  the  value  of  5  for  alternators  in  Table  60,  and  if  the 
displacement  in  electrical  degrees  is  to  be  the  same  for  all  alternators 
running  at  a  given  speed,  5  must  vary  inversely  as  the  number  of  poles,  or 
as  the  number  of  cycles  per  second.  This  would  indicate  that  the  arbi- 
trary use  of  the  usual  values  of  6  for  all  sorts  of  electrical  machinery  is  in 
error. 

If  the  constant  in  (31)  is  carefully  chosen  (and  it  depends  upon  the 
crank-effort  diagram  and  is  as  reliable  as  e),  the  simple  method  of  wheel 
weight  determination  in  Par.  116  is  reliable  and  saves  much  time;  but  in 
new  work  it  is  usually  desirable  to  investigate  by  more  elaborate  means, 
and  errors  may  sometimes  be  caught  in  this  way;  it  is  for  this  reason  that 
the  displacement  method  is  given  in  this  book. 

119.  Application  to  Practice. — A  wheel  will  be  designed  to  regulate  an 
alternating-current  generator,  neglecting  any  flywheel  effect  of  the  gen- 
erator. The  engine  is  the  20  by  48  in.  Corliss  engine  of  Chap.  XII.  The 
turning-effort  diagram  calculated  in  Chap.  XVI  has  been  reproduced  in 


364  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Fig.  227  of  this  chapter  and  is  the  basis  of  calculation  for  the  methods  of 
both  Pars.  116  and  117. 

For  a  rim  speed  of  1  mile  per  minute  the  wheel  diameter  could  be  16.8 
ft.  Assume  it  to  be  16  ft.  in  diameter.  While  the  maximum  power  at 
which  the  alternator  is  expected  to  give  the  best  service  should  probably 
be  taken,  the  diagrams  have  all  been  worked  out  for  the  rated  horse- 
power of  450,  and  this  will  serve  as  an  application  of  the  methods.  The 
diagram  factor  was  omitted  in  plotting  the  indicator  diagrams,  but  this 
is  probably  on  the  safe  side. 

Weights  will  first  be  determined  from  the  method  of  Par.  116,  using 
Formula  (8)  with  the  constant  C;  as  the  speed  is  less  than  5280  ft.  per 
min.,  K  can  not  be  used.  The  value  C  in  line  8  of  Table  60  has  been 
used  for  electric  generators  for  direct  or  alternating  currents,  whether 
belt-driven  or  direct  connected.  This  gives: 
_  11,620,000,000  X  450 

162  X  1003  2°'5C 

Line  9  for  a  25-cycle  (30-pole)  generator  gives: 
30  X  387,500,000  X  450 

162  X  100"  2°'5C 

For  a  60-cycle  (72-pole)  generator,  line  9  gives: 
72X387,500,000X450 

162  X  1003 

From  this  it  is  apparent  that  if  line  8  had  been  used  for  a  60-cycle 
machine,  there  might  have  been  trouble. 

The  displacement  method  is  given  in  Par.  1  17.     The  areas  and  values  of 

Ms  and  Mv  are  given  in  Table  61.     The  length  of  the  diagrams  as  origi- 

nally plotted  was  12  in.  and  it  was  intended  to  have  the  height  about  3  in. 

As  the  steam  engine  is  a  2-cycle  engine,  the  time  of  one  cycle  is,  from 

(13): 


then: 

0.6 


n_ 

12  =  °-05' 
As  already  determined  for  plotting  the  turning-effort  diagram  : 

n  =  10,000  Ib. 

From  Table  61,  the  maximum  plus  value  of  Mv  is  980,  the  maximum 
minus  value  340,  and  their  sum  (not  algebraic)  1320.  For  a  height  of  3 
in.  the  scale  would  be: 


q  =  -      -  =  440;  or,  take  it  as  500. 


FLY  WHEELS  365 

In  the  same  manner  the  scale  of  Ms  is: 

k  =  30. 
The  maximum  ordinate  measured  from  the  mean  line  of  Fig.  231  is: 

yM  =  1.72  in. 
From  (27),  for  a  25-cycle  alternator: 


Taking  a  =  2.5,  and  as  R  =  2,  (26)  gives: 


From  (25),  at  the  crank  pin: 
30  V  1  72 

M  =     AMEQOA     =  885°      and      W  =  32'16  X  885°  =  285,000  Ib. 
(J.OOooZO 

From  (28),  the  weight  referred  to  the  rim  is: 

WR  =  285,000  X    ^-          2  =  17,800  Ib.  ' 


In  the  same  manner  for  a  60-cycle  machine  we  have  : 
20  X  60 
~TOO~ 

Of)    y    1    70 


120  X  60  _    •  2.5X2 

P  "      ~TOO~  SM      28.65  X  72  = 


21,250  and  F  =  32.16  X  21,250  =  683,000  Ib. 


WR  =  683,000  X      -:=  42,750  Ib. 


Tabulating  the  results  for  comparison,  taking  Ww  =  about  1.6  WB, 
gives  Table  62. 


TABLE  62 


Method 

WR 

Ww 

Formula  (8),  line  8  of  Table  60     

20  500 

33  000 

Line  9  of  Table  60  —  30  poles 

20  500 

33  000 

Line  9  of  Table  60  —  72  poles  
Displacement  method  —  30  poles 

49,000 
17,800 

80,000 
29  000 

Displacement  method  —  72  poles 

42,750 

70000 

No  allowance  having  been  made  in  the  displacement  method  for 
possible  discrepancies,  the  values  are  some  smaller  than  by  the  simpler 
method,  which  in  the  present  problem  may  be  said  to  give  equally  good 
results.  Allowance  is  made  for  assuming  the  radius  of  gyration  as  the 
radius  of  the  outside  of  the  rim. 


366  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

In  Fig.  230  the  height  of  the  diagram  is  2.7  in.  and  the  scale  is  q/M. 
Then  the  speed  fluctuation  of  the  crank  pin  in  ft.  per  sec.  is: 
For  the  25-cycle  machine, 

2.7X500      ni_0 
v*-Vl*    "8850-    =0-153' 
For  the  60-cycle  machine, 

2.7  X  500 


2"l      -2172-50 
The  mean  velocity  of  the  crank  pin  is: 


_  2irRN  _  TT  X  2  X  100  _  01 

"60"  "30" 

Then  the  coefficient  of  speed  fluctuation  is  : 

For  the  25-cycle,     5  = 


0.0637         1 
For  the  60-cycle,     d  =  —^- 

Equating  these  with  (31)  gives: 


4.58p 

It  is  very  clear  that  5  must  depend  upon  the  number  of  poles  when  the 
engine  is  to  drive  alternators  in  parallel. 

In  Par.  106,  Chap.  XVI,  attention  was  called  to  the  inequality  of  crank 
effort  for  the  two  ends  of  the  cylinder  due  to  the  reduction  of  effective 
piston  area  by  the  piston  rod,  although  these  diagrams  were  drawn  for 
equal  cut-off.  In  some  small  steam  engines  with  simple  gear,  the  head- 
end cut-off  is  longer  than  the  crank-end,  increasing  the  inequality. 
When  no  provision  is  made  for  equalizing  cut-off,  this  should  be  taken 
into  account  in  plotting  diagrams.  With  gears  which  will  permit  it,  mak- 
ing the  crank-end  cut-off  longer  will  probably  give  a  more  uniform  turning 
effort;  but  most  erecting  and  operating  engineers  would  probably  try 
to  obtain  equal  cut-offs,  so  it  is  safer  to  ignore  such  a  possible  improve- 
ment in  drawing  the  diagrams. 

A  flywheel  for  this  problem  will  be  designed  in  Chap.  XXX. 

References 

Determination   of  flywheels  to  keep  the  angular  variation  of  an  engine  within  a 
fixed  limit.     Trans.  A.S.M.E.,  vol.  22,  p.  955. 

Flywheel  capacity  for  engine-driven  alternators.     Trans.  A.S.M.E.,  vol.  24,  p.  98. 


CHAPTER  XIX 
REGULATION  DURING  CHANGE  OF  LOAD.     GOVERNORS 

120.  Introduction. — The  governor,  or  regulator,  is  a  device  for  con- 
trolling the  speed  of  heat  engines  and  other  motors.  It  is  therefore  closely 
allied  with  the  valve-gear  mechanism  and  it  is  sometimes  a  little  difficult 
to  locate  the  dividing  line  between  the  two.  Commercially,  some  small 
throttling  governors  are  all  there  is  of  valve  gear  as  well  as  the  valve,  and 
needs  only  to  be  connected  with  the  steam  line  and  belted  to  the  engine 
shaft. 

In  general  the  parts  directly  associated  with  the  forces  which  are  in 
equilibrium  at  a  certain  speed  (called  the  tachometer  by  Zeuner),  and  the 
stand  or  wheel  to  which  they  are  attached,  comprise  the  governor;  the 
same  governor  may  be  applied  to  various  governor  problems  and  in 
different  ways. 

There  are  two  general  methods  of  applying  the  governor  to  the  control 
of  speed; 

1.  The  independent  method,  in  which  the  only  connection  with  the 
engine  to  be  governed  is  that  required  to  turn  the  governor  shaft.     This 
method  may  be  direct,  when  there  is  a  positive  connection — sometimes 
links  and  levers — between  the  governor  and  the  valve  controlling  the 
working  fluid  or  fuel ;  or  indirect,  when  the  governor  directly  controls  the 
supply  of  fluid  to  a  steam  or  hydraulic  cylinder  or  the  current  to  sole- 
noids, which  in  turn  operate  the  main  controlling  valve  or  valves. 

2.  The  coordinate  method,  in  which  the  governor  coordinates  with 
valve  gear  operated  by 'eccentrics  or  cams  which  receive  their  motion  from 
the  engine  shaft  and  bear  a  positively  timed  relation  to  it. 

The  governor  and  all  parts  connected  with  the  independent  method 
will  be  treated  in  this  chapter;  the  valve  gear  and  its  relation  to  the 
governor  will  be  considered  in  Chap.  XX. 

Due  to  more  or  less  contradiction  of  terms  in  governor  nomenclature, 
and  the  fact  that  different  names  are  given  to  the  same  thing  by  different 
builders  and  writers,  the  schedule  of  notation  will  take  the  form  of  defi- 
nition to  some  extent,  making  reference  easy. 

Notation. 

C  =  centrifugal  force  in  pounds  of  governor  weights  when  governor  is 
in  perfect  equilibrium  at  a  certain  speed,  neglecting  friction  or  any 

367 


368  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

externally  applied  force.  This  IB  balanced  by  the  centripetal 
force  caused  by  springs  or  dead  weights  belonging  to  the  governor 
proper. 

F  =  external  resistance  in  pounds  due  to  valve-gear  adjusting  mech- 
anism, opposing  change  of  position  of  governor  parts  in  either 
direction,  thereby  increasing  or  decreasing  the  effective  centri- 
petal force  as  the  speed  is  increased  or  decreased  respectively. 
F  is  sometimes  called  the  speed  regulating  force.  It  is  measured 
at  the  point  of  attachment  to  the  valve-gear  controlling  mechan- 
ism, as  the  sliding  sleeve  of  the  fly-ball  governor.  F  may  also 
be  assumed  to  include  the  friction  of  the  governor  bearings, 
ra  =  the  total  movement  in  inches  of  the  part  on  which  the  force  F 
acts.  It  is  sometimes  called  the  stroke  or  the  sleeve  lift. 

Fm  —  the  measure  in  inch-pounds  of  the  ability  of  the  governor  to 
overcome  external  resistance,  or  to  do  work  on  the  valve-gear 
adjusting  mechanism  with  a  given  speed  variation.  This  assumes 
the  mean  value  of  F  and  is  called  the  work  capacity  with  a  given 
speed  variation.  It  is  also  known  as  the  power  of  the  governor 
but  this  is  a  misnomer  as  power  involves  the  time  element. 

AC  =  change  of  centrifugal  force  from  any  value  C,  required  to  over- 
come resistance  F  and  allow  the  governor  to  assume  another  posi- 
tion corresponding  to  a  new  load.  In  any  governor  with  inertia 
effect  this  aids  in  overcoming  F. 

e  =  the  ratio  of  AC  to  C  ( =  AC/C)  The  ratio  'F/e  is  sometimes  known 
as  the  energy  of  the  governor  (this  is  a  misnomer  as  F/e  is  a  force), 
and  Fm/e  as  the  total  work  capacity  in  inch-pounds.  F/e  is  the 
effect  on  the  point  where  F  is  measured,  of  the  total  centrifugal 
force  of  the  revolving  weights. 

N  =  r.p.m.  of  the  governor  weights  in  any  position  of  equilibrium 

corresponding  to  C. 
A7V  =  change  of  N  corresponding  to  AC. 

8  =  the  ratio  of  AN  toN  (=  AN/N).  This  is  called  the  speed  variation 
by  some  makers  of  governors.  It  is  also  sometimes  known  as  the 
coefficient  of  sensitiveness.  F  and  8  are  interdependent,  but  as 
F  in  a  working  governor  is  the  actual  resistance  to  be  overcome, 
€  and  d  must  depend  upon  it.  As  8  measures  either  increase  or 
decrease  in  speed,  the  total  range  of  variation — or  fluctuation  due 
to  F  is  the  sum  of  8  in  both  directions,  or  practically  25,  as  demon- 
strated in  Par.  130. 

Ni  =  minimum  value  of  N,  for  maximum  load  on  engine.  A  load  caus- 
ing a  lower  speed  is  out  of  control  of  the  governor. 


REGULATION  DURING  CHANGE  OF  LOAD  369 

Nz  =  maximum  value  of  N,  for  minimum  load  on  engine.     A  limit  is 

sometimes  assumed  at  zero  load. 
NM  =  mean  of  Ni  and  Nz  (=  (Nt  +  JVi)/2). 

v  =  coefficient  of  fluctuation  of  speed  ( =  (N2  —  Ni)/N).  This  is 
sometimes  called  the  coefficient  of  speed  regulation,  and  by  some 
writers  is  considered  as  the  measure  of  sensitiveness.  If  v  is 
too  small,  "it  will  react  with  the  variation  of  turning  effort  within 
the  cycle.  This  leads  to  a  restless  governor  play  known  as  'hunt- 
ing."1 This  is  also  true  of  6.  The  value  of  v  depends  upon 
governor  proportions  only. 

a  =  the  coefficient  of  speed  fluctuation  for  the  engine  cycle  dependent 

upon  flywheel  regulation,  for  any  given  load.     The  value  of  this 

coefficient  depends  upon  the  variation  of  turning  effort  and  the 

kinetic  energy  of  the  flywheel. 

W  =  weight  of  centrifugal  weight  in  pounds.     This  may  be  a  single 

weight  or  may  be  divided  into  two  or  more  parts. 
r  =  the  distance  in  inches  from  the  axis  of  rotation  to  the  center  of 
gravity  of  the  weight  W  for  any  position.  This  should  include, 
referred  to  the  center  of  W}  all  parts  having  a  centrifugal  action. 
If  subscripts  1  and  2  are  used,  they  correspond  to  the  same 
numbers  applied  to  N. 

Mc  =  centrifugal  moment  without  friction  or  external  resistance. 
MT  =  centripetal  moment  without  friction  or  external  resistance. 
MP  =  moment  of  the  force  F. 
P  =  force  on  spring  in  pounds. 
For  other  notation  see  diagrams. 

121.  Governor  Types. — The  names  most  commonly  applied  to  gover- 
nors used  on  modern  heat  engines  are: 

Centrifugal  governors. 
Inertia,  or  centrifugal-inertia  governors. 
Relay  governors. 
Safety,  or  over-speed  governors. 

These  will  be  defined  in  a  general  way  in  Pars.  122  to  127.  A  mathe- 
matical treatment  will  be  given  centrifugal  governors  in  Pars.  128  to  133, 
these  being  of  the  only  type  capable  of  satisfactory  analysis. 

Governors  of  the  different  types  selected  from  practice  will  be  illus- 
trated in  Pars.  135  to  139. 

122.  Centrifugal  Governors. — In  these  governors  the  centrifugal  force 
of  revolving  weights  is  balanced  at  a  certain  speed  by  centripetal  force  due 
to  dead  weights  or  springs,  or  both.     Two  simple  forms  are  shown  in 

24 


370 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Fig.  232.  In  Fig.  232-A,  if  W  is  the  weight  and  C  the  centrifugal  force  of 
one  ball,  the  centrifugal  moment  about  pivot  x  is  Ch ;  the  centripetal  moment 
about  the  same  point  is  Wr.  These  moments  must  be  equal  to  be  in 
equilibrium,  so  for  some  definite  speed. 


Likewise  for  Fig.  233-5. 


Ch  =  Wr. 


Cb  =  Pa 


where  P  is  the  spring  load  balanced  by  one  weight.     A  more  extended 
treatment  will  presently  be  given. 

Any  change  in  the  load  on  the  engine  tends  to  change  the  speed ;  this 
changes  the  centrifugal  force  of  the  revolving  weights,  causing  it  to  bal- 
ance the  centripetal  force  in  a  new  position,  at  a  slightly  different  speed. 
This  new  position  adjusts  the  valve  gear  (including  the  mechanism  of 
the  independent  method)  to  the  new  load. 


FIG.  232. 

Stability. — A  governor  is  stable  if  a  change  of  position  necessitates  a 
change  of  speed  in  order  to  maintain  equilibrium,  an  increase  of  speed 
always  being  accompanied  by  an  increase  of  centrifugal  moment,  which, 
in  turn,  involves  an  increase  in  the  radius  of  rotation  of  the  revolving 
weights. 

Isochronism. — In  an  isochronous  governor  the  centrifugal  and  centri- 
petal forces  are  in  equilibrium  for  all  positions  of  the  governor  weights 
at  the  same  speed.  If  the  load  changes,  the  slightest  hint  at  change  of 
speed  causes  the  revolving  weights  to  fly  to  a  new  position  (if  the  external 
resistance  to  be  overcome  is  relatively  small),  adjusting  the  valve  gear  to 
suit  the  new  load,  at  the  original  speed.  On  account  of  friction,  strict 
isochronism  is  not  possible,  and  a  governor  theoretically  designed  for 
such  is  usually  unsatisfactory  and  unreliable.  A  certain  amount  of 
stability  is  desirable  even  for  service  requiring  the  closest  regulation. 


REGULATION  DURING  CHANGE  OF  WAD  371 

Isochronism  is  the  limit  of  stability.  If  this  limit  is  passed  an  engine 
would  run  faster  with  heavy  than  with  light  load.  One  engine  builder 
states  that  their  governor  has  regulated  satisfactorily  when  so  adjusted; 
but  it  is  of  the  centrifugal-inertia  type  and  supplied  with  a  dashpot, 
without  which  such  adjustment  would  be  impossible. 

A  conical  pendulum  governor  is  shown  in  diagram  in  Fig.  232-A.  This 
is  sometimes  called  the  Watt  governor.  If  a  central  weight  is  added 
as  shown  dotted,  it  is  called  a  loaded  governor,  or  Porter  governor.  The 
pendulum  governor  must  always  be  a  vertical-spindle  governor.  Some- 
times a  spring  is  used  instead  of  the  central  weight,  but  the  gravity  effect 
of  the  revolving  weights  is  still  an  important  factor,  so  a  vertical  spindle 
must  be  used. 

A  spring  governor  is  shown  in  diagram  in  Fig.  232-B.  This  particular 
form  is  sometimes  known  as  the  Hartnell,  or  Wilson  Hartnell  governor. 
The  gravity  effect  of  the  weights  is  small  compared  with  the  spring  load 
and  this  type  may  be  placed  in  any  position.  In  fact,  when  in  a  hori- 
zontal position  there  is  absolutely  no  gravity  effect,  while  for  the  vertical 
position  there  is  a  slight  gravity  moment  except  when  the  weights  are 
exactly  in  line  vertically  with  the  pivots.  There  are  different  forms  of 
spring  governors,  some  of  which  will  be  shown  in  later  paragraphs. 

Shaft,  or  flywheel  governors  are  located  on  the  engine  shaft,  and 
must  always  be  spring  governors.  They  may  have  either  one  or  two 
weights. 

123.  Inertia,  or  Centrifugal-inertia  Governors. — About  the  earliest 
example  of  this  type  to  be  practically  applied  was  the  Rites  governor, 
which  has  been  adopted  by  various  builders  and  modified  in  some  cases 
to  suit  their  requirements.  They  are  usually,  though  not  always,  shaft 
governors. 

Revolving  masses  tend  to  maintain  a  uniform  speed  unless  acted  upon 
by  some  external  force.  If  a  weight  is  caused  to  revolve  around  a 
governor  shaft,  being  connected  with  the  shaft  by  a  system  of  linkage  in 
such  a  way  that  a  change  of  angular  relation  between  shaft  and  weight 
operates  upon  the  valve  gear,  it  is  obvious  that  any  change  in  speed  of 
the  shaft  will,  due  to  the  inertia  of  the  weight,  effect  changes  of  adjust- 
ment. As  force  is  the  product  of  mass  and  acceleration,  the  ability  of  the 
revolving  weight  to  overcome  the  resistance  due  to  the  valve  gear  and 
change  the  position  of  the  gear  depends  upon  its  weight  and  the  time 
in  which  the  change  of  angular  relation  between  shaft  and  weight  occurs. 

As  the  inertia  weight  fixes  no  speed  limits,  but  acts  only  upon  sudden 
change  of  speed,  it  is  always  used  in  conjunction  with  a  centrifugal 
governor. 


372 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  principle  of  the  Rites  inertia  governor  may  be  illustrated  by  Fig. 
233.  A  bar  containing  a  weight  at  each  end  is  pivoted  on  the  governor 
wheel  at  A.  The  eccentric  rod  is  attached  at  B  and  swings  through  the 
arc  of  a  circle  as  the  governor  makes  adjustments.  The  right  end  of  the 
weight  is  heavier  than  the  other,  being  practically  in  gravity  balance 
about  the  pivot  A.  The  center  of  gravity  is  therefore  at  F  As  the 
wheel  revolves,  the  centrifugal  force  C  of  the  entire  weight  produces  a 
centrifugal  moment  Cb  about  A.  A  spring  connects  the  fixed  point  E 
on  the  wheel  with  point  D  on  the  weight,  producing  a  centripetal  moment 
Pa  in  an  opposite  direction  (clockwise)  to  Cb,  exactly  as  in  the  centrifugal 
governor  described  in  Par.  122,  and  at  a  certain  speed: 

Cb  =  Pa. 


FIG.  233. 

These  forces  are  not  great  enough  in  some  governors  of  this  type  to 
overcome  the  resistance  of  the  valve  gear  without  too  great  speed  varia- 
tion, but  the  necessary  force  is  furnished  by  inertia  I  of  the  weight.  If 
the  wheel  is  turning  clockwise  as  shown  by  the  arrow,  and  should  suddenly 
increase  in  speed  as  load  is  thrown  off  the  engine,  the  wheel  would  plunge 
ahead.  The  weight,  due  to  its  inertia,  would  tend  to  revolve  uniformly, 
the  inertia  /  acting  counterclockwise  as  indicated  in  Fig.  233.  The  in- 
crease in  centrifugal  force  due  to  increase  in  speed  would  tend  to  move  the 
weight  in  the  same  way  about  pivot  A.  Should  the  wheel  decrease  in 
speed,  both  inertia  and  centrifugal  force  would  act  to  move  the  weight 
in  a  clockwise  direction  relative  to  the  wheel. 

There  is  some  inertia  effect  at  some  positions  in  many  centrifugal  gov- 


REGULATION  DURING  CHANGE  OF  LOAD 


373 


ernors,  especially  of  the  shaft-governor  type.  When  present  the  work 
capacity  of  the  governor  is  determined  by  the  sum  of  the  centrifugal  and 
inertia  effects  if  they  are  in  the  same  direction,  and  by  their  difference 
if  opposed.  This  may  be  illustrated  by  Fig.  234.  Assume  a  wheel  to  be 
rotating  in  the  direction  R  (clockwise)  shown  by  the  full-line  arrow.  Let 
a  weight  be  pivoted  at  x,  and  assume  that  the  centrifugal  force  C  of  the 
weight  is  just  balanced  by  the  spring,  holding  the  weight  in  the  position 
shown,  at  a  certain  speed.  Now  assume  the  wheel  to  suddenly  increase 
in  speed.  The  centrifugal  force  of  the  weight  will  be  increased;  the 
weight  will  fly  out  until  the  tension  of  the  spring  balances  it  in  a  new  posi- 
tion. During  the  sudden  change  of  speed  the  weight  "hung  back" 
due  to  its  inertia,  exerting  the  force  /.  While  the  centrifugal  force  exerted 
a  positive  or  clockwise  moment  about  pivot  x,  the  inertia  exerted  a 


FIG.  234. 


FIG,  235. 


negative  or    counterclockwise    moment.     The  total  moment  is  then: 

M  =  Crc  -  Iru 

If  the  centrifugal  moment  is  greater  the  weight  would  go  outward,  making 
the  proper  adjustment  of  the  valve  gear;  but  if  the  inertia  moment  is 
greater  the  reverse  would  be  true. 

Suppose  the  governor  weight  to  be  pivoted  at  y  as  shown  dotted. 
This  causes  the  moment  of  /  to  act  in  the  same  direction  as  that  of  (7; 
then: 

M  =  Crc  +  IrH 

If  the  wheel  turns  counterclockwise  the  moments  are  in  the  same 
direction  for  the  full-line  position  of  the  weight,  and  opposed  for  the 
dotted  position.  It  is  clear  that  the  effect  of  /  will  be  zero  when  the  lines 
connecting  weight  center  with  wheel  center  and  with  pivot  form  an  angle 
of  90  degrees  as  shown  in  Fig.  235.  It  is  known  from  geometry  that  in 


374 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


this  position  the  weight  center  will  fall  on  a  circle  whose  diameter  is  a 
line  joining  the  wheel  center  and  pivot.  Then  from  what  has  preceded, 
if  the  effect  of  inertia  and  centrifugal  force  are  to  work  together  in  gear 
adjustments,  the  weight  should  always  remain  inside  the  circle  for  clock- 
wise motion  of  the  wheel,  and  outside  the  circle  for  counterclockwise 
motion.  This  is  shown  in  Fig.  236. 

In  some  governors  the  inertia  effect  is  nearly  negligible  and  such  are 
known  as  centrifugal  governors.  If  the  inertia  effect  is  great  they  are 
called  inertia,  of  centrifugal-inertia  governors. 

Inertia  governors  act  quickly  and  have  been  found  to  give  exception- 
ally good  regulation  under  widely  varying  and  suddenly  changing  loads, 
although  there  are  some  prominent  engine  builders  who  do  not  favor  much 
inertia  effect.  It  is  obvious  that  when  inertia  is  present  in  any  governor, 
its  moment  must  be  of  the  same  sign  as  the  centrifugal  moment  to  obtain 
the  best  results.  However,  all  governors  are  not  so  built. 


FIG.  236. 

Due  to  the  impossibility  of  predicting  the  relative  angular  accelera- 
tion of  shaft  and  inertia  weight  during  change  of  load,  the  design  of  in- 
ertia governors  is  dependent  upon  the  results  of  practice,  and  no  attempt 
will  be  made  to  treat  it  mathematically. 

124.  Relay   Governors. — When    considerable   force   is   required   to 
make  adjustments  of  the  valve  gear  a  hydraulic  cylinder,  usually  operated 
under  oil  pressure,  is  employed.     The  controlling  valve  of  the  oil  cylinder 
is  a  small  balanced  piston  valve  requiring  a  very  small  force  to  move  it; 
this  in  turn  is  controlled  by  a  governor  of  ordinary  capacity. 

125.  Safety,  or  Over-speed  Governors. — These  are  auxiliary  gover- 
nors used  in  addition  to  the  regular  governor,  more  usually  on  the  steam 
turbine.     They  consist  of  weights  and  spring  so  proportioned  that  at  a 
certain  excess  of  speed  the  weight  will  overcome  the  spring  tension  and 
operate  a  trigger,  release  a  heavy  weight  or  strong  spring  which  shuts  the 
throttle  valve  and  stops  the  turbine.     In  some  cases  the  relay  principle 


REGULATION  DURING  CHANGE  OF  LOAD  375 

is  used,  the  trip  mechanism  admitting  or  releasing  steam  from  an  auxiliary 
cylinder  whose  piston  is  connected  with  the  valve.  The  safety  governor 
is  a  simple  centrifugal  governor  and  is  adjustable. 

126.  Dashpots. — Under  certain  load  conditions,  especially  with  ex- 
tremely  sensitive   governors,    there    is   a   tendency   toward  a  restless 
movement  of  the  governor  weights,  sometimes  periodic  and  sometimes 
accompanying  load   changes.     This   action   is   called   hunting,    and   is 
sometimes  prevented  by  the  use  of  dashpots.     A  dashpot  consists  of  a 
cylinder  containing  a  fluid  such  as  oil,  and  supplied  with  a  piston  which 
allows  the  fluid  to  pass  between  it  and  the  cylinder  from  one  side  of  the 
piston  to  the  other  as  displacement  occurs.     The  piston  rod  is  attached 
to  some  moving  part  such  as  the  governor-sleeve  crosshead,  and  yields 
readily  under  gradual  changes,  but  offers  considerable  resistance  to  a 
sudden    change.     Sometimes   less    clearance   is    allowed    between    the 
piston  and  cylinder  walls,  the  connection  from  one  side  of  the  piston  to  the 
other  being  through  a  small  pipe;  a  valve  is  placed  in  the  pipe  so  that  the 
resistance  may  be  adjusted. 

Dashpots  may  be  seen  on  some  of  the  governors  illustrated  in  later 
paragraphs. 

127.  Speed  Adjustments. — The  speed  of  a  governor,   and  conse- 
quently the  speed  of  the  engine,  may  be  changed  by  changing  the  weight 
(except  in  the  simple  Watt  governor)  or  spring.     With  some  governors 
this  may  be  done  while  the  engine  is  running.     Where  several  engines  are 
direct-connected  to  alternators  running  in  parallel,  small  motors  operated 
from   the   switch-board — sometimes   automatically — are  in  some  cases 
attached  to  the  governors  in  such  a  way  that  the  tension  of  the  spring 
may  be  varied  and  the  alternators  brought  into  synchronism. 

128.  Speed  Fluctuation. — By  mean  speed,  the  speed  at  rated  load  is 
usually  meant,  and  this  is  usually  assumed  to  be : 

-       ,       .:!  ^-*+^      •  '  (1) 


in  which  N2  and  N\  are  maximum  and  minimum  r.p.m.  respectively, 
the  engine  is  designed  to  run  under  the  control  of  the  governor — possibly 
from  no  load  to  maximum  load. 

The  coefficient  of  the  fluctuation  of  speed  is: 

fa         '  .    |    „  =  ^^>  =  ?  <g-3_£)      .       '      ,  •  (2) 

If  v  is  assumed,  we  have  from  (1)  and  (2): 

(3) 


376  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and 

«  (4) 

If  a  governor  is  designed  with  this  range  of  speed  between  maximum 
and  zero  load,  it  is  not  at  all  certain,  or  perhaps  even  likely,  that  at  the 
position  the  governor  will  assume  in  running  the  speed  NM  the  gear  will 
be  adjusted  to  carry  the  rated  load;  but  this  is  not  essential,  and  Equa- 
tions (3)  and  (4)  are  convenient  in  the  design  of  spring  governors. 

Speed  fluctuation  and  the  coefficient  of  speed  fluctuation  depend  upon 
the  construction  of  the  governor  and  the  relation  of  the  forces  and  not 
upon  the  magnitude;  they  are  independent  of  friction.  It  is  true  that 
the  actual  fluctuation  may  be  greater  than  that  given  by  (2),  due  to 
friction,  but  this  is  considered  in  Pars.  129  and  133. 

129.  Speed  Variation.  —  The  centrifugal  force  of  a  revolving  weight  in 
Ib.is: 

WrN2 
n  _     nriy  /c\ 

35,230 

where  W  is  the  weight  in  lb.,  N  the  r.p.m.  and  r  the  radius  of  revolution 
in  inches,  of  the  center  of  gravity  of  the  weight. 

At  a  change  of  load  the  speed  changes,  increasing  for  decrease  of  load 
and  vice  versa.  However,  before  the  governor  can  change,  a  certain 
amount  of  external  resistance  F,  due  to  friction  and  weight  of  valve-gear 
parts,  has  to  be  overcome,  so  the  speed  changes  an  amount  A7V  before  C 
changes  enough  from  normal  to  overcome  the  resistance  and  move  the 
governor.  The  change  of  centrifugal  force  is  AC,  and  as  this  change  is 
made  before  the  governor  makes  its  adjustment,  it  occurs  at  a  constant 
value  of  r.  Then,  for  this  change,  it  may  be  seen  that  in  (5),  C  varies 
directly  as  TV2.  Let  the  speed  variation  be  denoted  by  8  ;  then  : 

The  ratio  of  correspondfng  values  of  centrifugal  force  is,  for  an 
increase  in  speed  : 


=  AC       (N  +  AJV2)2  -  N2       2N-&N  +  An2 
C 


= 

C  N2  N2 


Completing  the  quadratic  and  solving  for  8  gives: 


8  =  V*  +  1  ~  1  (8) 


REGULATION  DURING  CHANGE  OF  WAD  377 

Or,  approximately: 

«.=  J  +  i-i-|  (9) 

This  may  also  be  obtained  from  (7)  by  neglecting  the  minute  quantity. 
For  a  decrease  in  speed  : 

AC      N2  -  (N  -  AAO2      2AAT       /AA/\2 


=  25  -  d2 
From  which: 


_ 

6  =  1-       r^~e  (11) 

Or,  approximately  as  before  : 

;'•-';;     **i    :.;;  /  "VV:  ;•- 

Then  approximately: 

€  =  25  .  £  (12) 

In  practice  e  ranges  from  0.02  to  0.05.     Taking  the  largest  value: 

From  (7),  e  =  0.10  +  0.0025  =  0.1025 
From  (10),  e  =  0.10  -  0.0025  =  0.0975 
From  (12),  e  =  0.10. 

It  is  obvious  that  (12)  is  as  accurate  as  practice  will  warrant. 

The  total  range  of  variation  of  speed  without  adjustment  is  obviously 
26. 

130.  General  Equations  for  Centrifugal  Governors.  —  For  perfect 
equilibrium  at  any  speed  the  centrifugal  moment  must  equal  the  centri- 
petal moment  ;  or,  neglecting  friction  : 

Mc  =  MT  (13) 

At  the  instant  of  change  of  relative  position  of  the  governor  parts,  the 
moment  of  the  external  resistance  F  must  equal  the  fraction  e  of  the  mo- 
ment of  the  centrifugal  force  C  before  the  change  of  load  begins:  or: 

eMc  =  MF  (14) 

From  (13)  and  (14)  : 

MT  =  MF  (15) 

Equations  (13)  to  (15)  are  general  and  apply  to  all  centrifugal  govern- 
ors. From  (15)  it  is  obvious  that  if  €  is  to  be  constant,  MF/MT  must  be 
constant.  But  M  T  must  increase  for  light  engine  loads.  If  the  external 
resistance  F  is  constant,  the  moment  arm  of  F  must  increase  relative  to 
the  moment  arm  of  the  centripetal  force  (or  centrifugal  force).  Usually 
e,  and  consequently  6,  varies  with  change  of  position  when  F  is  uniform, 


378 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIG.  237. 


but  sometimes  a  compensating  device  is  employed  as  in  the  Jahns 
governor  described  by  means  of  Fig.  251. 

131.  Equations  for  Conical  Governors. — As  all  general  principles  were 
deduced  from  some  specific  case,  so  it  is  necessary  to  explain  them  in  the 
same  way.  For  the  purpose  the  equation  of  centrifugal  and  centripetal 
moments  was  illustrated  by  the  simple  sketches  of  Fig.  232.  The  equa- 
tion is  then  given  a  general  form  by  (13). 

It  is  quite  common  in  elementary 
works  dealing  with  governors  to  sub- 
stitute the  value  of  C  in  the  equation 
for  the  simple  Watt  governor  with 
arms  pivoted  at  the  center  line  of 
spindle  (Fig.  232-4),  and  show 
thereby  that  the  r.p.m.  is  propor- 
tional inversely  to  the  square  root  of 
the  height  h.  As  this  often  clings  to 
the  mind  as  a  general  rule,  such 
a  treatment  has  been  purposely 
avoided  and  a  more  general  method 

employed.  This  method  covers  the  simple  type,  but  includes  factors 
which  modify  results  considerably  when  this  type  is  not  strictly  adhered 
to,  as  it  seldom  is. 

Fig.  237  shows  a  conical  governor  with  central  weight  in  two  extreme 
positions;  the  full  lines  for  maximum  load  and  lowest  speed,  the  dotted 
lines  for  highest  speed  and  lightest  load  (if  this  is  taken  as  zero  load  it  is 
an  imaginary  position,  as  the  steam  or  fuel  supply  would  be  entirely  cut 
off).  This  is  known  as  the  Porter,  or  loaded  governor.  Fig.  237  is  made 
to  include  everything  for  the  most  general  treatment  of  this  type.  Should 
the  center  weight  not  be  used,  the  symbol  for  this  may  be  taken  as  zero 
in  the  final  equations.  When  this  weight  is  used  it  adds  to  the  centripetal 
force  only.  To  offset  this  the  centrifugal  force  must  be  increased  to  keep 
the  governor  parts  in  the  same  relative  positions.  As  increasing  the 
weight  of  the  revolving  weights  affects  centripetal  and  centrifugal  forces 
equally,  it  may  be  seen  from  (5)  that  the  speed  must  then  be  increased. 
The  revolving  weights  are  usually  made  smaller  when  the  center  weight 
is  used. 

In  deriving  the  equations  one  revolving  weight  only  need  be  consid- 
ered. The  component  Wo  of  the  center  weight  We  acting  on  one  side 
may  be  determined,  also  the  component  WF  of  external  resistance  F. 

Fig.  238  is  a  diagram  of  one  side  of  a  pendulum  governor  containing 
notation  used  in  the  equations.  The  center  weight  is  omitted  from  the 


REGULATION  DURING  CHANGE  OF  LOAD 


379 


sketch  but  included  in  the  calculations.  At  the  right  are  shown  force 
diagrams  for  finding  components  W0  and  WF  of  the  center  weight  and 
resistance  respectively.  The  arrows  indicate  the  direction  of  the  forces 
C,  W  and  Wc  ±  F.  The  last  shows  that  the  effective  centripetal  force 
at  the  sleeve  is  increased  as  the  governor  rises,  requiring  a  centrifugal 
force  greater  than  C  (by  the  amount  AC  =  eC)  to  overcome  it  before"  the 
governor  begins  to  rise;  and  on  descending,  the  effective  centripetal  force 
is  reduced  by  the  amount  F,  requiring  a  centrifugal  force  less  than  C 
(by  the  amount  eC)  before  the  weights  can  begin  to  descend  (se,e  Par.  129). 
Determination  of  Weights. — In  practical  design  it  is  necessary  to  first 
determine  the  weights  required  to  keep  the  speed  variation  d  (=  €/2) 


FIG.  238. 

within  the  desired  limit,  then  the  linkage  may  be  arranged  to  give  room 
for  these  weights. 

The  centripetal  moment  is  due  to  both  central  and  revolving  weights; 
then  from  Fig.  238,  for  one  side  of  the  governor,  taking  moments  about 
x,  the  point  of  suspension  of  the  arm: 

MT  =  WrP  +  W0r0  (16) 

Also: 

Mr  =  WFTO  (17) 

Substituting  (16)  and  (17)  in  (15)  gives: 

*(W  rP  +  Wore)  =  WFr0  (18) 

Equation  (18)  may  be  used  for  solving  governor  problems  by  trial 

and  error;  but  while  the  design  of  a  governor  is  largely  a  drawing  board 


380  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

problem,  the  work  may  be  shortened  by  making  a  number  of  substitutions 
in  (18).     Referring  to  Fig.  238,  let: 

a  +  b  .      Wc       , 


,  ,    /          tan  0\    sin  <£ 

rP  =  hp  tan  6  and  r0  =  hP  {  I    +  r  —  )         —  • 

\          tan<£/ 


From  the  force  diagrams: 


rrr  Wc  kW  ,         T,r  F 

W0  =  s-  —  =  in  7-7      and      W F  = 


2  cos  <f>      2  cos  <f>  2  cos 

Substituting  these  values  in  (18)  gives: 


1   tan 
or  simplifying: 

F 


2q 


-  =W    tan* 


(19) 


from  which  W,  F  or  e  may  be  found  if  the  others  are  known  or  assumed- 

In  practice  k  may  be  taken  from  10  to  15,  and  q  ranges  from  1  to  2,  a 
common  value  being  from  1.2  to  1.35. 

If  <f>  =  6  for  all  positions  the  equation  is  much  simplified,  but  advan- 
tage is  taken  of  a  difference  in  the  values  to  reduce  the  speed  fluctuation 
(see  Par.  128)  .  The  value  of  F  depends  upon  the  type  of  gear  and  size  of 
engine.  It  may  sometimes  be  determined  on  an  engine  already  built  by 
hooking  on  a  spring  balance  and  moving  the  parts.  For  a  small  or  me- 
dium-sized Corliss  engine  it  may  be  3  or  4  Ib.  As  previously  stated, 
e  (  =  26)  ranges  from  0.02  to  0.05,  depending  upon  the  desired  sensitive- 
ness; the  smaller  the  value  of  e  the  more  sensitive  is  the  governor. 

From  given  values  of  F  and  e,  it  may  be  seen  from  (19)  that  W  will 
have  the  greatest  value  when  tan  0/tan  6  (or  <£/0)  is  a  maximum.  Then  if 
c  is  taken  as  the  maximum  desired  value,  W  should  be  computed  for  the 
position  when  <f>/Q  is  maximum. 

Determination  of  Speed.  —  Solving  for  N  in  (5)  gives: 

N  =-  187.7^  (20) 

From  (13),  (16)  and  Fig.  238: 

ChP  =  WrP 
from  which: 


Wr'         °°-  (21) 


REGULATION  DURING  CHANGE  OF  LOAD 


381 


Observing  that  r  =  h  tan  0,  and  substituting  the  values  previously 
found  in  (21),  and  (21)  in  (20),  gives: 


N  =  187.7 


(22) 


FIG.  239. 


It  is  obvious  that  N  increases  as 
h  decreases;  if  tan  <£/tan  8  de- 
creases when  h  decreases  and  vice 
versa,  it  is  obvious  that  the  speed 
will  change  less  for  a  given  change 
of  the  governor.  This  means  that 
to  reduce  speed  fluctuation,  <£/0 
should  have  a  maximum  value 
when  h  is  maximum,  which  is 
when  the  governor  is  in  its  lowest 
position  (for  all  practical  gov- 
ernors). It  is  in  this  position, 
then,  that  W  should  be  deter- 
mined from  (19).  To  obtain  this 
result,  link  I  must  be  longer  than  a. 
If  a  spring  is  added  to  the 
weight,  or  entirely  takes  the  place  of  it,  k  will  be  variable,  but  equa- 
tions (19)  and  (22)  may  still  be  used.  If  the  spring  is  attached  in  any 
other  way  than  to  act  on  the  sliding  sleeve  as  does  the  center  weight,  the 
equation  of  Par.  130  should  be  used. 

It  is  clear  from  (22)  that  if  k  is  increased  as  the  governor  weights  de- 
scend and  decreased  as  they  rise,  N  will  vary  less.     In  governors  already 

built,  a  little  closer  regulation  may 
be  obtained  by  the  addition  of  some 
compensating  device.  In  Fig.  239, 
if  an  extension  be  added  to  the  bell 
crank  which  operates  the  cam  rods 
on  a  Corliss  Engine,  and  a  chain 
attached  to  this  and  to  a  fixed  point 
A  as  shown,  the  chain  will  exert  an 
upward  pressure  on  the  weight  when 
the  governor  is  in  the  lower  position 
(shown  dotted).  The  chain  may  lie  on  a  flat  support  instead  of  being 
suspended  from  A,  and  in  this  way  is  more  effective.  The  effect  at  the 
center  weight  support  is  inversely  proportional  to  the  length  of  lever  arm, 


382  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and  this  may  be  subtracted  from  Wc  and  the  values  of  k  found  for  the 
extreme  positions  and  N  found  from  (22). 

Proll's  governor  is  a  modification  of  the  pendulum  governor  having  the 
weight  on  the  lower  arm,  and  so  arranged  that  its  effect  is  similar  to  the 
chain  of  Fig.  239.  A  diagram  is  shown  in  Fig.  240.  The  effect  of  the 
revolving  weight  may  be  referred  to  the  juncture  of  the  links  and  to  the 
connection  of  the  lower  link  with  the  sleeve,  and  their  moments  taken 
about  x  as  for  the  Porter  governor.  The  effect  of  the  centrifugal  force 
of  the  ball  so  referred  is: 

r  =hn  WrBN* 
hA  35,230  ' 
Then: 

hBhP  WrBN* 
Mc  " "  T5T3 

The  effective  central  weight  is: 


Then: 

MT  =  WOETo  +  '-^W  (24) 

And: 

MF  =  WPr0 
as  before. 

Speed  variation  is  satisfactory  if: 


N  may  be  determined  by  equating  (23)  and  (24)  and  solving;  or: 


N  =  187'7  \  (WoETo  +  F)  (26) 

\  WhBhPrB  rA 

WCE  and  WOE  replace  Wc  and  W0  respectively  on  the  force  diagram  of 
Fig.  238. 

By  assuming  q  =  hB/hA,  W  may  be  solved  tentatively  by  (19).  The 
value  of  k  may  be  taken  as  for  the  Porter  governor;  the  larger  this  is,  the 
greater  the  effect  of  offsetting  We  in  the  high  position  of  the  governor, 
and  the  more  nearly  constant  the  speed.  This  governor  is  sometimes 
called  an  isochronous  governor  in  manufacturers'  catalogues. 

132.  Equations  for  Spring  Governors. — Fig.  241  shows  a  spring  gov- 
ernor which  may  be  used  to  derive  the  equations.  The  weight  of  W  is 
for  both,  or  all  of  the  revolving  weights  if  a  single  spring  is  employed,  as 


REGULATION  DURING  CHANGE  OF  WAD 


383 


is  usually  the  case  with  governors  of  the  type  shown  in  Fig.  241.  If 
separate  springs  are  used  as  in  some  shaft  governors,  W  may  be  for  one 
weight,  and  F  the  part  of  the  resistance  to  be  handled  by  one  weight. 
If  two  opposed  weights  are  connected  to  a  common  spring,  so  that  the 
deflection  is  double  what  it  would  be  with  a  separate  spring  for  each 
weight,  the  calculation  for  deflection  and  length  must  be  for  a  spring 
one-half  the  length  of  the  actual  spring; 
the  strength  calculations  are  based  upon 
one  weight. 
From  (13): 

Cb  =  Pa. 

It  is  more  general  to  assume  F  to  act  at' 
a  different  lever  arm  from  P;  call  this  a0 
Then  from  (15): 

e  Pa  =  FaQ. 
From  which : 


a0 


(27) 


FIG.  241. 


If  the  ratio  aQ/a  is  constant,  as  it  usually  is,  P  should  be  calculated  for 
the  least  spring  tension  in  order  that  e  may  not  be  less  than  the  assumed 
value;  then: 


Then  for  this  position: 


p1  = 


or: 


We  may  find 


61        35,230 
from  (3)  and  N2  from  (4).     Then  solving  for  W  gives : 


w  =  35,230Pi  01  =  35,230 


(28) 


Also: 


35,230    «2      *  ^asriW 
By  assuming  F/e  constant  in  (27),  adding  subscripts  1  and  2  for 
extreme  positions,  then  solving  for  F/e  and  equating,  we  have: 

(30) 

If  a0i  was  assumed  in  (27)  when  solving  for  PI,  a02  may  be  found.     This 
relation  provides  constant  speed  variation  d  for  a  constant  value  of  F, 


384 


DESIGtf  AND  CONSTRUCTION  OF  HEAT  ENGINES 


or  a  constant  ratio  F/e.     A  special  case  will  be  treated  under  the  Jahns 
governor  illustrated  in  Fig.  251,  Par.  135. 

Springs. — Forces  PI  and  P2  give  the  minimum  and  maximum  spring 
loads  required  for  a  desired  speed  fluctuation.  It  now  remains  to  select 
a  spring  which  will  give  these  values  with  a  change  of  deflection  equal  to 
m.  For  this  purpose  the  diagram  for  helical  springs  in  Fig.  242  is  useful. 
The  notation  on  the  sketch  is  clear,  all  dimensions  being  in  inches.  In 
addition  to  this: 


__/  engf-h  under 

_  o\      _  _  ^J^f£^ 

"Free  Spring  \~ 


O- — 

'onSprin^^\ 


Tens/ on 


'Free  Spring*  I 

y        Length  uncter_  I 

""~Max.Loc*ot"~ 


FIG.  242. 

Es  =  torsional  modulus  of  elasticity  of  spring  wire. 
S  =  maximum  allowable  fiber  stress  in  spring  wire. 
n  =  number  of  coils  in  the  spring. 
PM  =  the  maximum  safe  load  in  pounds. 

The  initial  deflection  i  is  required  to  produce  load  PI,  and  the  addi- 
tional deflection  m  requires  the  load  PI.  From  Fig.  242,  by  similar  tri- 
angles : 

m  +  i  =Pz 
i          Pi 
From  which: 

i--~  (31) 


-- 
Pi 

From  a  spring  table,  given  in  engineering  handbooks,  select  a  spring 


REGULATION  DURING  CHANGE  OF  LOAD 


385 


with  a  safe  load  Ps  ^  P2,  and  which  will  deflect  the  amount  i  with  the 
load  Pi,  or  i  +  m  with  load  P2. 

In  the  absence  of  spring  tables  the  following  formulas  may  be  used: 


PM   = 


and: 


I  = 


8dc*nPl 


An  empirical  formula  for  safe  stress  is: 

10,000 

o   <£  — 3 h  oU,UUU 

«ir 

Not  to  exceed  100,000 


(32) 


(33) 


(34) 


Laminated  springs  are  used  in  some  governors,  and  while  they  are 
satisfactory  and  the  theory  is  simple,  it  is  not  quite  so  easy  to  predict 
results  as  when  helical  springs  are  used.  The  formulas  are  based  upon 
the  bending  of  a  triangular  plate  which  has  been  cut  up  into  leaves  as 
shown  in  Fig.  243,  in  which  the  dimensions  are  T 
in  inches,  and  n  the  number  of  plates.  The 
theoretical  formula  for  maximum  safe  load  is: 

^  (35) 


PM 

and  for  deflection : 


Enbt 


(36) 


FIG.  243. 


where  E  is  the  direct  modulus  of  elasticity.  The 
value  of  E  for  steel  is  29  to  30  million,  but  Good- 
man says  that  26  million  should  be  used  in 
calculation;  the  difference  is  probably  from  deflec- 
tion due  to  shear,  and  the  fact  that  the  small 
central  plate  is  left  out.  In  practical  springs  the  ends  of  the  leaves  are 
squared  as  shown  dotted.  There  is  often  more  than  one  full-length  leaf, 
and  when  the  number  of  such  leaves  is  one-fourth  the  entire  number,  5.5 
is  sometimes  used  instead  of  6  in  (36).  Friction  also  alters  the  deflection 
of  laminated  springs;  if  well  oiled  the  discrepancy  is  not  great. 

Sometimes  a  plate  spring  is  used  which  curves  nearly  around  the  wheel 
as  in  Fig.  244.  The  application  of  force  P  as  shown  by  the  arrow  produces 
a  bending  moment  at  any  point  in  the  spring  equal  to  Px.  If  the  free  end 
of  the  spring  were  guided  in  the  direction  of  application  as  indicated,  the 
bending  moment  at  a  would  be  zero.  It  does  not  deflect  in  this  way  how- 
ever, and  the  elastic  curve  is  very  complicated.  The  spring  may  be 

25 


386 


DEMON  AND  CONSTRUCTION  OF  HEAT  ENGINES 


checked  for  strength  by  (35),  using  values  of  x  instead  of  /;  rough  deflec- 
tion calculation  may  be  made,  but  the  design  of  such  springs  must  be 
largely  empirical. 

Condition  of  Stability. — For  any  position  of  the  governor: 

b    WrN2 

=  rN2  X  constant 


P  = 


a  35,230 


or: 


N  = 


p 

-  X  constant. 
r 


If  P/r  is  constant  the  governor  is  isochronous. 

If  P  varies  in  the  same  direction  as  r  but  at  a 
greater  rate,  N  varies  in  the  same  direction  as  P 
and  the  governor  is  stable. 

If  P  varies  in  the  same  direction  as  r  but  at  a 
lesser  rate,  N  varies  in  the  opposite  direction  from 
P  and  the  governor  is  unstable. 

Some  governors  are  so  constructed  that  the 
moment  arms  for  centrifugal  and  centripetal  force 
vary  in  different  ways  throughout  the  range  of 
governor  action,  and  it  is  often  wise  to  plot  a 
curve  showing  these  relations. 
Practical  Considerations. — To  obtain  the  best  results  in  practice, 


FIG.  244. 


133. 


the  speed  fluctuation  of  the  governor  should  be  greater  than  the  speed 
fluctuation  of  the  cycle  allowed  by  the  flywheel;  otherwise  there  will  be 
governor  action  or  hunting  during  every  revolution  of  the  wheel.  This 
is  also  true  of  the  fluctuation  of  speed  due  to  change  of  load,  or  26. 

The  speed  fluctuation  should  never  be  less  than  twice  the  speed  varia- 
tion and  should  usually  be  greater.  These  relations  may  be  expressed 
thus: 

v  >  a 

25  >  a  (37) 

v  ^  26 

Weight  of  Links. — When  heavy  weights  or  strong  springs  are  employed, 
the  weight  of  arms  and  links  may  usually  be  neglected.  It  is  more 
accurate  to  include  them  and  this  may  be  easily  done.  In  getting  their 
centripetal  moment,  their  weight  may  be  assumed  concentrated  at  their 
center  of  gravity  but  for  their  centrifugal  moment  this  would  not  be 
correct.  For  the  upper  link,  dividing  the  link  into  a  number  of  parts, 
the  centrifugal  moment  is: 

M     -  s  wrN*  tttt 

Mc"S35^30'a 


REGULATION  DURING  CHANGE  OF  LOAD 


387 


where  w  is  the  weight  of  one  part  (see  Fig.  245).  For  the  lower  link,  the 
effect  must  be  transferred  to  the  upper  link  at  the  juncture  of  the  two; 
then: 


wrN2    x    • 
35,230  'T 


(39) 


Gravity  Balance.— As  far  as  possible,  all  parts  of  a  governor  should  be 
in  gravity  balance  for  all  parts  of  the  revolution.  This  applies  especially 
to  governors  with  horizontal  shafts. 

Selection  of  Constants  of  Regulation. — Goodman  says:  "If  a  governor 
be  required  to  work  over  a  very  wide  range  of  power,  such  as  all  the  load 
suddenly  thrown  off,  a  sensitive,  almost  isochronous 
governor  with  dash-pots  gives  the  best  results;  but 
if  very  fine  governing  be  required  over  small  vari- 
ations of  load,  a  slightly  less  sensitive  governor 
without  dash-pots  will  be  the  best."  He  further 
says:  " Nearly  all  governor  failures  are  due  to 
their  lack  of  power."  The  word  power  as  used 
here  is  a  misnomer,  meaning  the  product  Fmt  the 
work  capacity  with  a  given  speed  variation.  It  is 
therefore  safe  to  assume  F  amply  large;  on  the 
other  hand,  if  the  actual  value  falls  too  far  short 
of  the  assumed  value,  d  is  reduced,  and  the  rela- 
tions given  by  (37)  may  not  obtain. 

Machine  Design. — Failure  of  a  governor  due  to  broken  parts  or 
loosened  bolts  is  exceedingly  serious,  and  great  care  should  be  exercised 
in  design  and  construction.  Friction  of  joints  and  bearings  should  be 
eliminated  as  much  as  possible  and  ample  lubrication  provided  for.  All 
governors  should  be  thoroughly  tested  and  adjusted  before  leaving  the 
factory. 

ILLUSTRATIONS  FROM  PRACTICE 

134.  Weight-loaded  Conical  Governors. — A  good  example  of  the 
loaded  governor  of  the  Porter  type  is  given  in  Fig.  246,  which  is  used  on 
the  Bass  Corliss  engine  built  by  the  Bass  Foundry  and  Machine  Co., 
Fort  Wayne,  Ind.  In  this  governor,  part  of  the  center  weight  is  hollow 
to  receive  shot  for  adjusting  the  weight.  A  sliding  weight  is  also  placed 
on  the  bell  crank  to  allow  for  further  adjustment.  A  dashpot  is  furnished 
to  prevent  wide  fluctuation  when  the  load  is  changed. 

A  safety  pawl  is  provided  which  prevents  the  governor  falling  to  its 
lowest  position  when  starting  the  engine.  It  is  so  constructed  that  when 
the  engine  gains  sufficient  speed  to  lift  the  sleeve,  the  pawl  falls  by  grav- 


388 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


ity;  then  should  the  governor  belt  break  or  slip,  the  governor  upon  coming 
to  rest  may  sink  to  its  lowest  position,  in  which  it  engages  the  safety 
cams  and  prevents  the  engine  from  taking  steam.  The  so-called  safety 
collar  which  must  be  turned  to  the  safety  position  by  hand  after  the 
engine  has  started,  is  more  convenient  when  starting  the  engine,  but  is 
only  a  partial  safety  device;  neglect  to  set  the  collar  may  result  in  disaster. 


FIG.  246. — Bass-Corliss  governor. 

The  Bass  engine  is  sometimes  fitted  with  an  idler  pulley  running  on 
the  governor  belt.  This  connects  with  the  pawl,  and  in  case  of  belt 
accident,  falls,  pulling  the  pawl  out  of  position.  This  has  the  advantage 
of  easy  starting  and  stopping.  This  device  is  shown  on  a  governor  about 
to  be  described. 

A  governor  of  the  Proll  type,  built  by  the  Nordberg  Manufacturing 
Co.  for  use  on  their  Corliss  engines  is  shown  in  section  in  Fig.  247- A,  and 


REGULATION  DURING  CHANGE  OF  LOAD 


389 


a  side  elevation  in  Fig.  247-.B.  Fig.  247-J5  is  arranged  for  a  simple  engine 
of  moderate  size  having  the  Nordberg  long-range  cut-off  gear.  It  is 
provided  with  a  dashpot  and  has  an  idler  pulley  which  operates  the  safety 
device.  In  a  portion  of  the  connection  between  the  governor  crosshead 
and  the  double  bell  crank,  a  spring  may  be  seen  holding  two  parts  to- 
gether. Between  these  two  parts  is  a  separating  strut  with  a  projecting 
lever  containing  a  pin  which  slides  in  a  slot  of  the  rod  which  is  fastened 


A  B 

FIG.  247. — Nordberg  governor. 

to  the  idler  arm.  Should  the  belt  fail  or  run  off,  the  idler  would  drop; 
the  end  of  the  slot  would  strike  the  pin,  remove  the  strut  and  allow  the 
spring  to  raise  the  bell-crank  lever,  which  for  this  type  of  gear  prevents 
steam  admission,  and  the  engine  would  stop. 

A  Nordberg  governor  of  the  Porter  type  is  shown  in  Fig.  248.  This  is 
employed  when  the  engine  is  used  to  drive  an  alternator  in  parallel.  It 
is  also  designed  for  the  long-range  gear,  but  the  details  of  design  are 
different.  A  small  motor,  shown  at  the  right,  is  provided;  this  is  operated 


390 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


from  the  switch  board,  and  by  driving  a  lead  screw,  shifts  a  counter- 
weight, varying  the  speed  and  bringing  the  engine  into  synchronism.  A 
spring  is  interposed  between  the  gear-operating  lever  and  the  dashpot, 
and  the  safety  latch  releases  a  weight  if  the  belt  breaks,  which  forces  the 
levers  in  the  safety  position. 


FIG.  248. — Nordberg  governor. 

135.  Spring-loaded  Flyball  Governors. — Fig.  249  shows  the  simple 
spring  governor  used  on  the  smaller  sizes  of  ''Economy"  turbines  built 
by  the  Kerr  Turbine  Co.,  Wellsville,  N.  Y.  It  is  located  on  the  end  of 
the  turbine  shaft,  its  connection  with  the  governor  throttle  valve  being 
shown  in  Fig.  250.  The  governor  weights  408  are  of  semi-annular  cross- 


REGULATION  DURING  CHANGE  OF  LOAD 


391 


,,•401 


27 


FIG.  249. — Kerr  turbine  governor. 


FIG.  250. — Kerr  governor  on  turbine. 


392 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


section,  fitted  with  hardened  tool-steel  knife  edges  402  which  are  securely 
fitted  into  the  governor  cup  400.  The  governor  weights  have  rolling 
contacts  shown  by  the  dotted  lines,  and  act  against  the  governor  collar 
404,  transmitting  their  motion  to  it.  The  governor  collar  in  turn  trans- 
mits its  motion  to  the  governor  spindle  411  through  the  spindle  pin  406. 
The  spindle  has  a  sliding  clearance  in  the  reamed  hole  provided  for  it  in 
the  end  of  the  turbine  shaft.  A  governor  spring  410  adjusted  by  nut 
415  works  against  the  governor  weights  through  the  spring  sleeve  409 
and  the  governor  collar.  A  stop  nut  412  screwed  up  to  such  a  position 
that  the  governor  spring  sleeve  comes  out  against  it  at  its  extreme  outer 
travel,  prevents  the  weights  from  throwing  out  of  their  seats.  The 
guard  401  is  fastened  to  the  governor  cup,  serving  to  protect  the  weights 
and  supplementing  the  stop  nut  in  preventing  over-travel  of  the  governor 
weights.  Any  movement  of  the  governor  is  transmitted  to  the  governor 
valve  through  the  pivot  block  424  and  the  governor  lever  425.  The 
thrust  comes  on  a  race  of  ball  bearings  428.  Lock  nut  427  holds  the  ball 

race  in  place.  Lubrication  is  supplied 
by  an  oil  cup  426.  Small  cord  pack- 
ing keeps  the  oil  from  running  out 
along  the  spindle.  This  packing  is 
kept  in  place  by  gland  430  and  nut 
431. 

In  some  governors  of  the  Jahns 
type,  the  movement  of  the  sliding 
sleeve  is  effected  by  bell  cranks  en- 
gaging with  the  sleeves  through  roller- 
bearing  connections,  with  races  set  at 
such  angles  that  the  movement  of  the 
weights  due  to  centrifugal  force  is 
transmitted  in  such  a  manner  that 
the  force  F  exerted  by  the  sleeve  is 
practically  constant  for  each  position 
of  its  travel.  This  was  mentioned  in 
Par.  130,  and  may  be  illustrated  by 
Fig.  251  which  is  somewhat  exaggerated  for  the  purpose.  Using  the 
same  notation  as  in  Par.  132,  remembering  that  lever  arm  a  is  coincident 
with  b,  and  taking  d  as  the  apparent  arm  for  F,  i't  is  clear  fhat : 


<.. J 


FIG.  251. 


CL2 


But  due  to  the  slanting  roller  race,  arm  a0  is  shorter  for  all  positions 


REGULATION  DURING  CHANGE  OF  LOAD  393 

by  the  constant  amount  c.     It  is  then  clear  that  the  ratio  of  arm  a0  to 
arm  a  is  greater  in  position  2  than  in  position  1;  or: 


FIG.  252.  —  Kerr  turbine  governor. 

To  have  e  and  F  constant,  or  c  vary  as  F,  the  following  relation  should 
hold: 


az(di  -  c)       Pi 


394 


or:- 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 
a02      P^2 


(40) 


which  is  the  same  as  (30). 

This  governor  is  used  to  quite  an  extent  on  steam  turbines  and  inter- 
nal-combustion engines. 

Fig.  252  shows  the  governor  used  on  the  larger  turbines  built  by  the 
Kerr  Turbine  Co.,  and  is  used  in  connection  with  an  oil  relay  system  of 
governing.  It  is  driven  from  the  turbine  shaft  by  spiral  gears.  Below 
the  speed  governor  the  safety  governor  A  is  located;  this  will  be  de- 


FIG.  253. — Mclntosh  &  Seymour  governor. 

scribed  later.  At  the  lower  end  of  the  governor  is  a  gear  pump  which 
supplies  the  oil  relay  cylinder  and  the  lubricating  system.  The  supply 
line  is  connected  to  the  pump  discharge  line  ahead  of  a  30-lb.  relief  valve. 
The  oil  supplied  to  the  bearings  is  taken  from  between  the  discharge 
from  the  30-lb.  relief  valve  and  a  3-lb.  relief  valve.  This  provides  30  Ib. 
pressure  for  the  relay  cylinder  and  3  Ib.  for  the  lubricating  system. 
Except  in  detail  of  design  this  governor  is  practically  the  same  as  the 
Kerr  governor  described  at  length  earlier  in  this  paragraph.  The  pivot 
block  B  which  operates  the  governor  lever  is  at  the  top  of  the  spindle. 
The  governor  yoke  C  fitted  with  hardened  tool-steel  knife-edge  blocks 
carries  on  it  two  governor  weight-arms  D.  These  compress  the  spring 
by  means  of  two  pivot  struts. 


REGULATION  DURING  CHANGE  OF  LOAD 


395 


136.  Centrifugal  Shaft  Governor. — Fig.  253  shows  the  shaft  governor 
built  by  the  Mclntosh  and  Seymour  Corporation,  Auburn,  N.  Y.  and  used 
on    their    steam   engines.      It   is    of    the 

centrifugal  type.  To  quote  from  Bulletin 
No.  52  of  their  Type  F  engines:  "The 
governor  is  so  designed  that  the  com- 
bined inertia  effect  of  the  different  parts 
due  to  change  of  speed,  while  it  acts  in 
the  proper  direction  to  assist  the  governor, 
is  very  slight,  as  a  considerable  inertia 
effect  has  proved  to  result  in  instability 
and  unsatisfactory  governing."  It  will 
be  noticed  that  this  is  contrary  to  the 
views  of  other  builders  who  use  inertia 
governors. 

The  governing  eccentric  is  placed  upon 
a  fixed  eccentric  in  such  a  way  that  the 
radius  of  the  eccentric  path  and  the 
angular  position  of  the  eccentric  relative 
to  the  crank  are  both  changed  as  the 
governor  weight  changes  position.  A  leaf 
spring  is  used  to  balance  the  centrifugal 
force  of  the  weight  at  the  desired  speed. 

137.  Inertia,     or     Centrifugal-inertia 
Governors. — Fig.  254  shows  the  centrifu- 
gal-inertia governor  of  the  Lentz  engine 
built  by  the  Erie  City  Iron  Works,  Erie, 
Pa.     The    hub    D   is   keyed    to   the   lay 
shaft  which  is  geared  to  the  engine  shaft. 
The  centrifugal  weights  A  are  pivoted  to 
this    hub.     They  are   also  connected  by 
links    E    to  the  inertia   weight  B.     The 
spring  C  connects  the  inertia  weight  with 
an    arm    on    hub    D.     The   governor   as 
shown  in  Fig.  254  turns  counterclockwise. 
Should  the  load  decrease  and  the  engine 
speed  up,  the  hub  D  tends  to  accelerate 
in    a    counterclockwise    direction.     Cen- 
trifugal force  tends  to  throw  the  weights 

out;  the  inertia  weight  lags  and  has  the  same  action  on  the  weight 
levers. 


396 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


A  hand  wheel  on  the  end  of  the  shaft  moves  the  rod  G,  on  the  end  of 
which  is  a  taper.  This  raises  or  lowers  the  rod  F  which  presses  upon  the 
spring  near  its  support  and  changes  the  tension.  Thus  the  speed  of  the 
engine  may  be  changed  while  it  is  running.  This  may  be  operated  by  a 
small  motor  from  the  switchboard  when  the  engine  is  driving  an  alternator 
in  parallel. 


FIG.  255. — Kerr  turbine  relay  governor. 

138.  Relay  Governors. — The  Kerr  Turbine  Co.  relay  governor  is 
shown  in  Fig.  255.  The  pilot  valve  is  shown  at  591.  Oil  under  pressure 
enters  at  the  center  of  the  valve  chamber  and  is  admitted  to  one  end  or 
the  other  of  oil  cylinder  539.  The  exhaust  from  the  oil  cylinder  passes  out 
near  the  ends  of  the  valve  chamber  into  the  space  surrounding  the  driv- 


REGULATION  DURING  CHANGE  OF  LOAD       .  397 

ing  gears.  The  governor  is  shown  in  its  extreme  highest  position  with  the 
governor  valve  556  closed.  This  valve  is  a  double-seated  balanced 
valve  and  opens  downward.  Upon  starting  it  is  in  its  extreme  lowest 
position. 

The  following  description  is  taken  from  Bulletin  26  of  the  Kerr 
Turbine  Co. : 

"As  the  turbine  is  started  up  the  weights  are  held  in  position  by  the 
governor  spring  until  centrifugal  force  begins  to  compress  the  spring.  As  the 
speed  increases  the  weights  swing  outward  in  a  larger  and  larger  circle  trans- 
mitting their  motion  to  the  governor  lever  and  giving  it  an  upward  motion. 
This  causes  an  upward  motion  of  the  relay  pilot  valve  591  and  uncovers  thp 
ports  so  as  to  release  the  pressure  on  the  upper  side  of  the  oil  relay  piston  540 
and  admit  pressure  on  the  bottom  side.  This  causes  an  upward  travel  of  the 
piston  and  decreases  the  steam  valve  opening.  At  the  same  time  the  motion 
of  the  valve  is  communicated  to  the  governor  lever  560  through  the  starting 
lever  548,  which  is  pivoted  at  549  and  the  starting  lever  link  rod  541.  This 
downward  travel  of  the  lever  at  its  outer  end  brings  the  oil  relay  pilot  valve 
back  toward  its  original  position,  gradually  decreasing  the  flow  of  oil,  and 
finally  a  position  is  reached  where  the  steam  valve  opening  admits  the  proper 
amount  of  steam  for  the  load  to  be  handled.  The  pilot  valve  will  then  be  in 
a  central  position  and  the  pressure  on  the  top  and  bottom  of  the  relay  piston 
will  be  equal.  When  the  speed  decreases,  the  governor  weights  travel  in 
toward  the  governor  shaft,  causing  a  downward  motion  of  the  governor  lever 
and  pilot  valve  which  opens  up  the  chamber  below  the  piston  to  exhaust  and 
the  chamber  above  the  piston  to  admission.  This  causing  downward  travel 
of  the  steam  valve,  admitting  more  steam,  at  the  same  time  causing  an  upward 
movement,  of  the  governor  lever,  which  brings  the  pilot  valve  back  toward  its 
original  position  until  the  speed  and  load  become  settled  again.  The  pilot 
valve  will  now  be  in  a  central  position  and  the  pressure  on  the1  top  and  bottom 
of  the  relay  piston  will  be  equal. 

"To  receive  close  regulation  the  starting  lever  link  rod  541  is  moved 
toward  the  fulcrum  549  of  the  starting  lever;  and  for  wide  speed  regulation  and 
great  stability  it  is  pivoted  toward  the  outer  end  of  the  starting  lever.  The  relay 
pilot  valve  will  always  run  with  the  ports  all  closed  as  soon  as  the  load  is  settled ; 
in  other  words,  the  relay  pilot  valve  is  the  fulcrum  of  the  governor  mechanism. 
A  speed  variation  of  from  2  to  5  per  cent,  can  be  secured  while  the  turbine  is 
running  by  shortening  or  lengthening  the  starting  lever  link  rod  541,  which  has 
right-  and  left-hand  threads  on  the  ends.  This  speed  variation  can  also  be  ob- 
tained on  all  turbines  driving  alternators  by  means  of  the  synchronizer.  This 
consists  of  a  synchronizer  spring  case  1582  to  which  is  attached  the  upper  and 
lower  covers.  The  upper  one  has  a  clearance  hole  through  which  the  stem, 
which  has  a  collar  attached,  can  move.  The  upper  end  is  connected  to  a  continu- 
ation of  the  governor  lever.  By  means  of  the  adjusting  wheel  the  spring  may  be 


398  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


*- To  H.P.Lim 

FIG.  256. — Kerr  turbine  mixed-pressure  valve. 


REGULATION  DURING  CHANGE  OF  LOAD 


399 


given  any  degree  of  compression  desired,  and  locked  there  by  means  of  the  locking 
wheel.  The  greater  the  compression  of  the  spring,  the  greater  the  speed  of  the 
turbine.  A  speed  of  about  5  per  cent,  above  that  given  by  the  regular  governor 
can  be  obtained." 

The  steam  valve  shown  in  Fig.  255  is  the  regular  high-pressure  valve. 
A  mixed-pressure  valve  is  shown  in  Fig.  256.  The  starting  lever  fulcrum 
is  shown  at  H.  The  top  valve  is  the  low-pressure  valve  admitting  steam 
to  the  low-pressure  element  on  the  turbine  shaft.  The  bottom  valve 
admits  steam  to  the  high-pressure  element.  When  there  is  sufficient 


FIG.  257. — Allis-Chalmers  safety  governor. 

low-pressure  steam  to  carry  the  load,  the  operation  is  exactly  as  described 
for  the  regular  high-pressure  turbine;  the  valve  stem  then  slides  through 
the  high-pressure  valve  without  engaging  with  it.  Should  the  low-pres- 
sure steam  not  be  ample  for  the  load,  the  slight  decrease  in  speed  causes 
the  governor  weights  to  take  a  new  position  toward  the  center,  lowering 
the  pilot  valve  and  admitting  pressure  to  the  top  side  of  the  oil  piston. 
The  nut  B  then  engages  with  the  high-pressure  valve,  forcing  it  open 
against  the  spring,  which  forces  the  valve  again  to  its  seat  when  the  stem 
ascends.  The  turbine  now  operates  as  a  mixed-pressure  turbine,  or  if 
there  is  no  low-pressure  steam,  as  a  high-pressure  turbine.  A  check 
valve  in  the  low-pressure  steam  line  prevents  an  outflow  of  steam  from 
the  turbine,  should  the  pressure  exceed  that  in  the  low-pressure  main. 
A  small  piston  on  the  lower  end  of  the  valve  stem  is  connected  with  the 
high-pressure  steam  line  for  the  purpose  of  balancing  the  valves,  oil  piston 
and  stem. 


400 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


139.  Safety,  or  Over-speed  Governors. — Fig.  257  shows  the  safety 
governor  of  the  Allis-Chalmers  Co.  It  is  located  in  the  end  of  the  tur- 
bine shaft  and  does  not  depend  upon  gears.  It  is  a  simple  spring  gover- 
nor, which,  when  some  predetermined  speed  is  reached,  compresses  the 
spring,  forces  out  the  spindle  and  releases  the  trigger  A .  This  releases 
the  rod  B,  to  the  end  of  which  a  heavy  weight  is  hung.  This  weight  in 
turn  is  attached  to  a  lever  which,  when  the  weight  drops,  closes  the 
main  throttle  valve  and  stops  the  turbine. 


FIG.  258. — Bruce-Macbeth  mixing  valve. 

140.  Gas  engine  governors  are  usually  of  the  spring-loaded  type.  In 
some  cases  they  coordinate  with  the  valve  gear,  and  such  arrangements 
will  be  treated  in  Chap.  XX.  More  commonly  they  operate  the  throttle 
valve,  which  is  usually  a  small  butterfly  valve  in  constant-volume  oil 
engines.  In  gas  engines  the  throttle  valve  is  known  as  a  governor 
valve  or  mixing  valve.  Such  a  valve,  used  on  the  Bruce-Macbeth  engine 
for  natural  and  illuminating  gas  is  shown  in  Fig.  258. 


CHAPTER  XX 
VALVES  AND  VALVE  GEARS 

141.  Introduction. — The  design  of  valve  gears  is  essentially  a  drawing 
board  proposition.  The  division  of  mechanics  most  used  is  kinematics, 
It  is  true  that  gear  parts  must  have  sufficient  strength  and  wearing  surf  ace, 
and  the  analysis  to  determine  forces  acting  is  not  usually  very  difficult; 
but  the  force  required  to  move  the  valve  is  uncertain  in  many  cases  and 
assumptions  must  be  made  which  makes  the  method  largely  a  rule  of 
thumb. 

Corliss  valves  in  large  engines,  though  unbalanced,  are  easily  moved 
by  hand  so  long  as  they  are  kept  in  motion,  showing  that  the  film  of 
steam  between  valve  and  seat  partly  balances  them.  By  assuming  poor 
lubrication,  and  taking  the  force  required  to  move  the  valve  after  it  has 
stood  a  while  under  pressure,  forces  acting  on  all  parts  of  the  gear  are 
easily  found. 

There  is  no  force  theoretically  required  to  move  balanced  valves  if 
friction  is  neglected;  they  may  be  computed  the  same  as  unbalanced 
valves,  but  as  one  of  the  advantages  of  balanced  valves  is  the  reduction 
in  weight  of  the  gear,  a  certain  proportion — such  as  one-half — of  the 
force  may  be  used. 

The  details  illustrated  later  in  this  chapter  will  be  given  as  a  guide, 
but  the  computations  will  be  omitted  in  most  cases.  The  dimensions  are 
usually  the  result  of  experience,  and  the  stresses  and  bearing  pressures 
may  be  found  when  the  gear  layout  is  complete. 

The  problem  of  treating  the  subject  of  valve  gears  in  a  single  chapter 
is  difficult  and  may  only  be  done  with  any  degree  of  satisfaction  by  dis- 
cussing with  reasonable  thoroughness  a  few  of  the  gears  in  common  use 
which  involve  the  principles  common  to  other  gears,  illustrations  of  the 
latter  being  given  with  little  or  no  discussion. 

A  number  of  gears  are  in  use  today  on  engines  built  some  years  ago, 
but  these  gears  are  not  manufactured  today;  such  gears  are  then  practic- 
ally obsolete  from  the  standpoint  of  the  designer,  and  little  or  no  attention 
will  be  given  to  them.  It  may  be  thought  by  some  that  the  Corliss 
gear  is  entering  this  class,  but  as  this  was  practically  the  first  to  give  a 
shortened  valve  travel  with  quick  opening  and  is  still  used  in  both  the 

26  401 


402  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

older  and  modified  forms,  with  and  without  the  releasing  mechanism, 
it  is  treated  in  a  fairly  thorough  manner. 

No  special  valve  diagram  is  championed;  those  selected  will  enable 
nearly  eveiy  gear  problem  to  be  handled.  The  Zeuner  diagram,  while 
elegant,  may  be  used  for  no  problem  which  may  not  be  better  solved  by 
one  of  the  other  diagrams  given;  neither  does  it  clarify  the  principle  of 
valve  action  better;  therefore  it  is  omitted. 

For  a  larger  collection  of  gear  designs  and  a  more  thorough  treatment 
of  some  phases  of  the  subject  the  reader  is  referred  to  the  excellent  books 
mentioned  at  the  end  of  this  chapter. 

Notation. 

a  =  port  area. 

S  =  piston  speed.     Also  stress,  and  on  valve  diagrams,  steam  lap. 
A  =  piston  area. 
V  =  steam  or  gas  velocity. 
D  =  cylinder  diameter. 
w  =  width  of  port. 

I  =  length  of  port  =  kD.     Also  lead  on  valve  diagrams. 
d  =  poppet  valve  diameter. 
h  =  poppet  valve  lift. 
r  =  VC/VF. 
q  =  d/D. 


$,  P,  E,  T,  B,  W  and  I  are  dimensions  on  diagrams. 
A,  Q,  D,  L,  C,  R  and  M  are  angles  on  diagrams. 

142.  Port  Area.  —  About  the  first  requirement  in  valve  gear  design  is 
the  port  area.  This  is  determined  by  assuming  the  port  wide  open 
throughout  the  stroke,  and  that  the  steam  or  gas  is  of  constant  specific 
volume,  although  this  is  really  far  from  true,  especially  during  the  exhaust 
stroke.  But  with  this  assumption  the  areas  of  port  and  cylinder  bore 
are  inversely  proportional  to  mean  velocities  of  steam  or  gas  and  piston; 
or: 

aV  =  AS  (1) 

in  which  a  and  A  are  areas  in  sq.  in.  of  port  and  cylinder  bore  respectively, 
S  is  the  mean  piston  speed  in  ft.  per  min.  and  V  the  nominal  mean  gas 
or  steam  velocity  in  ft.  per  min.  Then  : 

AS  ,0. 

a  =    T 

The  length  of  the  port  in  steam  engines  (which  will  be  taken  as  its 
greater  dimension)  varies  from  about  0.8  the  cylinder  diameter  in  small, 


VALVES  AND  VALVE  GEARS  403 

slow-speed  steam  engines  to  somewhat  greater  than  the  cylinder  diameter 
for  locomotives  with  flat  valves.  Corliss  engines  usually  have  ports 
equal  in  length  to  the  cylinder  diameter.  For  piston  valves  the  length 
is  much  greater,  while  in  poppet  valves  the  area  is  equal  to  the  product  of 
lift  and  effective  circumference. 

As  a  general  guide,  it  is  considered  desirable  to  have  the  port  opening 
at  any  part  of  the  piston  stroke  proportional  to  the  piston  velocity  at 
that  point,  the  maximum  port  opening  corresponding  to  the  maximum 
piston  speed.;  it  is  obvious,  however,  that  this  could  be  possible  only 
with  engines  in  which  the  valve  is  open  during  the  entire  stroke  unless 
opening  and  closing  is  instantaneous,  but  it  gives  a  basis  of  comparison 
which  will  be  used  in  the  following  paragraphs. 

Allowable  Fluid  Velocity  V.  —  For  steam,  the  following  formulas  may  be 
used  as  a  guide.  These  are: 

Steam  port,       V  =  7000  ^l~  (3) 

Exhaust  port,    V  =  5000  ^l~  (4) 

By  varying  the  numerical  coefficient  these  formulas  may  be  used 
for  the  ports  of  internal-combustion  engines.  Values  for  these  engines 
will  presently  be  given  in  connection  with  poppet  valves. 

If  double  ports  are  used  each  opening  may  be  taken  one-half  of  the 
value  found  by  (2). 

By  the  term  port  opening,  the  amount  that  the  valve  uncovers  the 
port  is  meant.  In  single-valve  engines  where  the  same  ports  are  used 
for  inlet  and  exhaust,  the  port  is  made  large  enough  to  accommodate 
the  exhaust  steam;  then  the  inlet  edge  of  the  valve  does  not  usually 
uncover  the  entire  port.  Also,  due  to  certain  characteristics  of  the 
valve  motion  the  exhaust  edge  of  the  valve  usually  has  considerable 
over-travel.  The  terms  steam  and  exhaust  are  used  in  common  parlance 
for  the  incoming  and  exhaust  steam  respectively. 

Slide  valves  may  be  used  to  include  all  those  having  sliding  surfaces. 
The  relative  proportions  of  such  valves  and  their  seats  are  determined  by 
practice  with  the  various  kinds.  As  an  example  of  design,  double-ported 
Corliss  valves  will  be  taken.  Fig.  259  shows  sections  of  steam  and 
exhaust  valves  at  one  end  of  the  cylinder.  By  experience  with  the  layout 
of  Corliss  valve  diagrams  the  following  proportions  were  found  to  give 
good  results: 

w  =  i  (5)         &  =  o.5d   (6)         wi  =  0.233d   (7)         bi  =  0.6d   (8) 


c  =  0.27rf   (9) 


404 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Certain  other  dimensions  are  given  which  may  be  used  in  cylinder  design. 
The  values  of  b  and  61  were  made  ample  to  provide  sufficient  seal  when  the 
valves  are  closed;  this  may  often  be  made  less  if  desired  and  may  be 
determined  or  checked  by  means  of  the  valve  diagram  given  later. 

As  separate  valves  are  used,  the  ports  may  be  entirely  uncovered,  and 
there  is  no  necessity  of  excessive  over-travel  of  the  exhaust  valve.     Some 


1  Jfeam  Passage 


Exhaust  Passage 


FIG.  259.  —  Corliss  double-ported  valves. 


over-  travel  is  advantageous  as  it  gives  a  quicker  opening  and  allows  for 
adjustment.  The  valves  in  Fig.  259  are  shown  wide  open  (the  steam 
valve  moves  in  a  counterclockwise  direction  in  opening,  and  the  exhaust 
valve  clockwise),  with  some  over-travel,  also  some  clearance  at  the  heel 
of  the  valve  to  insure  against  partly  closing  the  second  port  in  case  of 
adjustment;  this  does  not  give  an  ideal  steam  passage,  but  is  apt  to  give 


VALVES  AND  VALVE  GEARS  405 

better  practical  results  than  too  little  allowance.  The  operation  of  this 
valve  will  be  better  understood  from  a  study  of  the  gear,  the  layout  of 
which  is  also  determined  experimentally  and  in  connection  with  valve 
and  port  design;  there  is  no  distinctly  logical  order  in  proceeding. 

The  length  of  the  port  is  usually  equal  to  the  cylinder  diameter  in 
Corliss  engines,  but  a  fraction  of  this  length  is  taken  up  by  ribs  which 
connect  the  bridge  with  the  two  parts  of  the  cylinder  for  strength  and 
stiffness  and  to  insure  against  distortion  due  to  heat.  The  length  may 
then  be  taken: 

I  =  kD 

where  k  may  be  from  0.85  to  0.9;  or  it  may  be  any  desired  fraction  for 
special  designs.  Then  the  area  of  the  steam  port  in  Fig.  259  is: 

a  =  2wl  =  2kwD 
The  piston  area  is: 

A  "    V 
Substituting  these  values  in  (2)  gives : 


Then  from  (5) : 


If  in  (11),  V  is  taken  as  the  inlet  velocity,  the  correct  velocity  will 
be  obtained  through  the  exhaust  port,  as  w\  from  (7)  is  proportioned  so 
that: 

Vw 


-8kV 


Having  determined  d,  other  dimensions  may  be  found  by  Formulas 
(6)  to  (9).  As  V  is  greater  for  larger  cylinders,  the  valve  diameters  are 
relatively  smaller.  In  some  large  low-pressure  cylinders  the  diameter  of 
the  steam  valves  is  less  than  of  the  exhaust  valves,  the  relative  proportions 
being  the  same. 

Other  types  of  valves  may  be  treated  in  the  same  general  way. 

Single-ported  poppet  valves  are  used  almost  entirely  for  internal-com- 
bustion engines.  Some  have  flat  seats  as  in  Fig.  260-  A,  but  usually  the 
seat  is  conical  with  an  angle  of  45  degrees  as  in  Fig.  260-#. 

The  area  of  opening  of  the  flat-seated  valve  is: 

aF  =  irdh  (12) 


406  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  area  through  a  conical  seat  is: 


i  =  ird  — 7=  + 


2V2 

The  subscripts  F  and  C  denote  flat  and  conical  seats  respectively. 
Let: 

Vc       aP  1.414 


r  = 


VF       ac        1+0.5^ 
a 


(13) 


(14) 


A  6 

FIG.  260. — Single-ported  poppet  valves. 

In  Table  63  are  values  of  r  for  the  complete  range  of  h/d  found  in  practice. 

TABLE  63 


h/d  ,. 

0.040 
1.386 

0.060 
1.373 

0.080 
1.360 

0.100 
1.345 

0.120 
1.333 

0.140 
1.321 

0.160 
1.310 

0.180 
1.297 

0.200 
1.285 

0.220 
1.273 

0.240 
1.262 

0.250 

1.257 

When  h/d  =  0.12,   near  the  center  of  the  range,  r  =  %.     For  this 
value  (14)  gives: 

dp  3  3     j,  /ir\ 

dc   ==   =    7  Q>F    =   77Ttt/l  (Id) 

r        4  4 

It  may  be  assumed  that  this  gives  the  area  of  opening  for  all  poppet 
valves.  It  is  on  the  side  of  safety  for  all  values  of  h/d  greater  than  0.12, 
and  these  values  are  most  common  in  modern  engines.  The  error  is 
slight  for  smaller  values.  The  theoretical  opening  is  greater  in  flat- 
seated  valves,  but  on  account  of  the  obstruction  to  flow  offered  by  sharp 
corners,  it  is  doubtful  if  the  capacity  is  greater  than  for  conical  valves 
having  the  same  lift.  The  conical  seat  is  usually  much  narrower  than 
shown  in  Fig.  260,  the  opening  approaching  that  of  the  flat  seat  for  high 
lifts.  The  lift  of  a  flat-seated  valve  to  give  full  opening  of  the  port  is 
J4  the  diameter  of  the  port,  neglecting  the  stem.  For  a  conical  seat 
it  is  higher,  but  d/4  is  the  maximum  value  of  h  given  in  treatises  on  inter- 
nal-combustion engines,  and  this  is  seldom  used. 

Dropping  the  subscript  in  (15)  and  substituting  in  (2)  gives: 

3 

4 


8. 


VALVES  AND  VALVE  GEARS 

Dividing  through  by  d*  gives: 

rr  h 


407 


Let  h/d  =  m  and  d/D  =  q;  then: 

V 

S 


1 


(16) 


3mq2 

From  (16)  Table  64  has  been  calculated.  This  table  will  be  found  useful 
in  preliminary  calculations,  and  for  any  assumed  or  measured  values  of 
m  and  q  the  value  of  V/S  is  seen.  This  factor  multiplied  by  S  gives  the 
nominal  gas  velocity. 

TABLE  64 


Q. 

Values  of  V/S  for  m  = 

0.06 

0.08 

0.10 

0.12 

0.14 

0.16 

0.18 

0.20 

0.22 

0.25 

0.25 

89.00 

66.60 

48.00 

44.50 

38.00 

33.30 

29.40 

26.70 

24.30 

21.40 

0.30 

61.80 

46.30 

37.00 

30.90 

26.50 

23.10 

20.60 

18.50 

16.80 

14.80 

0.35 

45.40 

34.00 

24.50 

22.70 

19.40 

17.00 

15.10 

13.70 

12.40 

10.90 

0.40 

34.70 

26.00 

18.80 

17.40 

14.90 

13.00 

11.60 

10.40 

9.44 

8.34 

0.45 

27.40 

20.60 

14.80 

13.70 

11.80 

10.30 

9.14 

8.23 

7.49 

6.59 

0.50 

22.20 

16.70 

12.00 

11.10 

9.50 

8.34 

7.40 

6.66 

6.06 

5.34 

0.55 

18.40 

13.80 

11.00 

9.18 

7.85 

6.89 

6.12 

5.50 

5.01 

4.40 

0.60 

15.50 

11.60 

9.26 

7.71 

6.60 

5.78 

5.14 

4.63 

4.20 

3.70 

0.65 

13.20 

9.88 

7.90 

6.57 

5.63 

4.93 

4.38 

3.94 

3.59 

3.16 

0.70 

11.40 

8.52 

6.80 

5.66 

4.85 

4.25 

3.78 

3.40 

3.09 

2.72 

0.75 

9.88 

7.42 

5.92 

4.94 

4.22 

3.70 

3.29 

2.96 

2.69 

2.37 

0.80 

8.70 

6.52 

5.21 

4.34 

3.73 

3.25 

2.89 

2.60 

2.37 

2.09 

0.85 

7.70 

5.78 

4.61 

3.84 

3.29 

2.88 

2.56 

2.30 

2.01 

0.90 

6.86 

5.15 

4.12 

3.43 

2.94 

2.57 

2.29 

2.05 

0.95 

6.15 

4.62 

3.69 

3.08 

2.64 

2.31 

2.05 

1.00 

5.55 

4.17 

3.33 

2.78 

2.38 

2.08 

Various  values  of  V  are  given  by  different  authorities.  Giildner 
recommends  4500  ft.  per  min.,  with  6000  as  a  maximum  under  favorable 
conditions;  but  these  values  do  not  seem  feasible  for  modern  high-speed 
engines.  It  is  true  that  the  lower  the  value  of  V/S  can  be  kept  the  greater 
will  be  the  capacity  of  the  engine,  therefore  as  large  values  of  m  and  q 
should  be  selected  as  is  practicable.  A  few  data  of  representative  engines 
are  given  in  Table  65. 

The  dimensions  of  the  airplane  engine  were  scaled  from  a  drawing  and 
may  not  be  strictly  correct.  If  the  piston  speed  were  2000  ft.  per  min., 
V  would  be  12,900.  The  low  value  of  V/S  in  airplane  engines  is  no 
doubt  an  important  factor  in  enabling  them  to  obtain  as  high  an  m.e.p. 
as  130  Ib. 


408  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  65 


Engine 

Valves 

m 

q 

v/s 

a 

V 

4-cylinder  truck      

inlet 

0  156 

0  448 

10  60 

1  196 

12  600 

4-cy  Under  truck              .  . 

exh 

0  171 

0  448 

9  70 

1  196 

11  530 

4-cylinder  auto 

inlet 

0  170 

0  440 

10  10 

1  660 

16  800 

4-cylinder  auto  

exh. 

0  208 

0  440 

8  25 

1  660 

13  700 

6-cylinder  auto  

both 

0  200 

0  433 

8  95 

1  565 

14  000 

6-cylinder  auto 

inlet 

0  182 

0  440 

9  42 

1  750 

16  500 

6-cylinder  auto  

exh. 

0  222 

0  440 

7  70 

1  750 

13500 

8-cylinder  airplane  

both 

0  227 

0  477 

6  45 

? 

? 

Double-ported  poppet  valves  are  used  for  superheated-steam  engines 
and  uniflow  engines.  The  port  area  may  be  taken  twice  that  of  the 
single  valve,  or: 

3     "  (17) 


a  =  —  Trdh 
F      _L 

s  = 


Also: 

Table  64  may  be  used  by  taking  one-half  the  tabular  value. 

143.  Classification. — The  term  valve  gears  is  practically  always  as- 


FIG.  261. 

sociated  with  reciprocating  engines,  as  in  these  there  is  always  a  complete 
cycle  of  operations.  This  is  not  true  of  the  steam  turbine;  the  gear  is 
not  in  time  relation  with  the  wheels,  but  is  closely  connected  with  the 
governor,  so  is  treated  in  Chap.  XIX. 

A  classification  of  the  gears  of  reciprocating  engines  is  difficult  and 
not  very  satisfactory,  but  lists  will  be  given  for  steam  and  internal  com- 
bustion engines;  by  dividing  into  a  number  of  headings  a  general  idea  of 
methods,  valves  and  gears  may  be  had,  and  these  terms  will  be  used  in 
paragraphs  which  follow. 

By  the  term  eccentric,  a  sort  of  enlarged  crank  pin  is  usually  meant, 


VALVES  AND  VALVE  GEARS 


409 


Valve  system 


Gear  drive 


Valve  control 


the  diameter  of  which  is  such  that  the  shaft  diameter  lies  within  it;  this 
is  so  for  the  reason  that  the  eccentric  is  a  casting  which  surrounds  the 
shaft  and  is  fastened  to  it  by  key  or  set  screw  or  both.  This  is  shown  in 
Fig.  261-.A.  The  center  of  the  eccentric  often  lies  within  the  shaft  circle 
as  shown.  On  certain  locomotive  valve  gears  and  some  shaft  governors, 
the  gear  is  not  driven  by  such  an  eccentric,  but  by  a  small  pin  as  shown 
in  Fig.  261-5.  This  is  also  known  as  the  eccentric  and  will  be  so  included 
when  the  term  is  used  in  this  book. 

(1)  Steam  engines. 

Single  valve 

Double  valves  (expansion  valve) 

Binary  or  separate  valves 

Uniflow 

Eccentric 

Eccentric  and  wrist  plate      nonreversing 

Eccentric  and  cam 

Eccentric 

Eccentric  and  crosshead       reversing 

Connecting  rod 

Shifting  eccentric 

Swinging  eccentric 

Releasing  gear 

Reversing  link  motion 

r  TT  ,    .          ,     I  Plain  D-valve 
I  Unbalanced    <-,.».  , 

Flat  J  {  Gridiron  valve 

[  Balanced.     Single-  to  triple-ported 
Piston.     Balanced,  single-  or  double-ported 
Rocking.     Unbalanced,  single-  to  quadruple-ported 
Poppet.     Balanced,  double-ported 
Piston  acting  as  valve 

(2)  Internal-combustion  engines. 

(  4-cycle.     Mechanically  operated  inlet  and  exhaust 

4-cycle.     Automatic  inlet  valve 
Valve  system  <    .        , 

|  4-cycle.     Sleeve  valve 

[  2-cycle.     2-port  and  3-port 
f  Cam 

Gear  drive..  .  {  Cam  and  eccentric 
[  Eccentric 

^i  -j.1.  jv     j.j.v  /  Inlet  valve 

Constant  gear  motion  with  throttling  governor  <  „    .      . 

Releasing  gear  for  inlet  or  fuel  valve 

Hit-and-miss 

Fuel  valve  or  pump  control 
f  Poppet 

Valves j  Sleeve 

[  Piston  acting  as  valve 


Valves. 


Valve  control 


410 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


144.  Single -valve  Gear. — This  is  the  simplest  form  of  gear  and  it 
sometimes  seems  the  most  difficult  to  understand,  as  all  events  are 
accomplished  for  both  ends  of  the  cylinder  by  a  single  valve.  The  subject 
is  usually  approached  by  a  consideration  of  the  valve  without  lap  or  lead, 
and  with  no  angular  advance  of  the  eccentric;  these  terms  will  be  ex- 
plained presently.  Angularity  of  connecting  rod  and  eccentric  rod  will 
also  be  neglected,  it  being  assumed  that  the  Scotch  yoke  is  used  for  each; 
then  the  displacement  of  the  valve  and  piston  is  measured  by  the  projec- 
tion upon  the  line  of  stroke  of  the  eccentric  center  and  crank  pin  center 
respectively. 

A  sectional  diagram  is  shown  in  Fig.  262  of  such  a  gear.  A  general 
description  of  steam  engine  operation  is  given  in  Chap.  Ill,  in  which  the 
cycle  is  traced  through  for  a  4-valve  engine,  so  it  will  not  be  necessary  to  go 


#A  I   Kfc 


FIG.  262. 

into  details  here.  In  Fig.  262,  a  is  the  steam  chest  and  b  the  exhaust 
passage  leading  from  the  cylinder.  The  crank  and  eccentric  circles  are 
shown  separately  to  avoid  confusion. 

The  piston  is  at  the  head  end  of  the  stroke  and  the  crank  at  the  head- 
end dead  center.  The  eccentric  is  at  right  angles  with  the  crank,  causing 
the  valve  to  be  in  the  center  of  its  travel,  just  covering  both  ports.  As 
the  eccentric  leads  the  crank,  a  movement  in  a  clockwise  direction  will 
open  the  head-end  port  to  the  steam  chest  a,  and  the  crank-end  port  to 
the  exhaust  passage  b.  If  the  movement  is  continued  until  the  crank 
is  at  the  crank-end  dead  center,  the  valve  will  again  be  at  its  central 
position  with  both  ports  closed.  Then  admission,  cut-off,  release  and 
compression  all  occur  at  the  end  of  the  stroke  and  the  indicator  diagram 
is  a  rectangle. 

The  arrangement  just  described  in  which  the  high-pressure  steam  is  in 
a  on  the  outside  of  the  valve,  gives  what  is  known  as  outside  admission. 
If  steam  entered  at  b  (necessitating  some  device  to  keep  the  valve  on  the 


VALVES  AND  VALVE  GEARS 


411 


seat)  and  left  past  the  outer  edges  of  the  valve,  entering  a  and  thence  to 
the  atmosphere,  the  arrangement  is  known  as  inside  admission;  then  to 
open  the  valve  at  the  head  end  to  high-pressure  steam,  the  crank  and 
eccentric  must  rotate  in  a  counterclockwise  direction,  and  this  is  the 
principle  of  operation  of  the  steering-gear  engine  used  on  steam  ships. 
Lap,  Lead  and  Angular  Advance.  Still  assuming  the  crank  to  be  at  the 
head-end  dead  center  and  that  there  is  outside  admission,  let  us  arbitrarily 
add  steam  lap  S  and  exhaust  lap  E  to  the  valve  as  in  Fig.  263-A,  and  study 
the  effect.  The  port  is  now  closed,  and  the  valve  must  be  moved  to  the 
right  the  distance  >S  (the  steam  lap)  before  it  begins  to  open,  and  the 


FIG.  263. 

eccentric  must  move  through  the  angle  Q  (the  lap  angle)  to  bring  the  valve 
to  the  opening  position,  or  admission.  If  the  eccentric  and  crank  are 
still  at  right  angles  as  in  Fig.  262,  the  crank  would  travel  the  angle  Q 
from  dead  center  before  admission  would  occur,  which  is  not  permissible. 
It  then  becomes  necessary  to  advance  the  eccentric  through  the  angle  Q 
while  the  crank  is  still  at  the  head-end  dead  center;  this  angle  is  now 
denoted  by  A,  as  it  is  the  angular  advance.  In  some  books  the  angular 
advance  is  the  angle  between  crank  and  eccentric  measured  from  the 
crank  in  the  direction  of  motion.  This  seems  logical,  but  it  is  American 
practice  to  consider  angular  advance  the  angle  between  eccentric  and 
crank  minus  90  degrees,  and  it  will  be  so  considered  here. 

If  the  eccentric  radius  were  such  as  to  just  open  the  valve  to  steam 


412  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

before  steam  lap  were  added,  the  opening  would  now  be  P  —  S,  in  which 
P  is  the  width  of  the  port.  It  is  usual  to  determine  this  opening  as  in 
Par.  142;  then  denoting  the  steam  port  opening  by  Ps,  the  eccentric 
radius  must  be  made  equal  to : 

\  =  S  +  Ps  (18) 

where  T  is  the  valve  travel,  which  is  twice  the  eccentric  radius  when  the 
eccentric  drives  the  valve  directly  as  assumed  in  this  discussion.  For- 
mula (18)  is  correct  in  any  event,  as  it  contains  only  valve  measurements. 
Admission  position  is  shown  in  Fig.  263-5.  As  explained  in  Chap.  Ill, 
it  is  customary  to  open  the  valve  to  steam  a  little  before  the  end  of  the 
stroke  is  reached ;  the  amount  the  valve  is  open  when  the  crank  is  on  dead 
center  is  called  lead,  and  is  denoted  by  I.  The  angle  the  eccentric  and 
crank  must  pass  through  to  move  the  valve  the  distance  I  is  called  the 
lead  angle.  The  angular  advance  is  the  algebraic  sum  of  the  lap  angle  and 
lead  angle]  or: 

A  =  Q  +  L  (19) 

Lead  is  shown  in  Fig.  263-C,  the  crank  (not  shown)  being  at  the 
head-end  dead  center. 

It  now  remains  to  follow  the  crank  from  admission  through  a  complete 
revolution,  tracing  events  and  determining  the  indicator  diagram  pro- 
duced. This  is  done  in  Fig.  264  for  the  head-end  port.  Arrows  show 
direction  of  rotation,  direction  of  piston  travel  (on  indicator  diagrams) 
and  direction  of  valve  travel.  Full  valve  sections  are  shown  at  the  top 
with  valve  in  central  position  (the  position  when  lap  is  measured)  and 
at  the  extreme  position  with  head-end  steam  port  and  crank-end  exhaust 
port  open.  In  the  crank-eccentric  diagram,  crank  and  eccentric  are 
shown  connected  by  a  light  line  to  avoid  confusion.  The  center  line 
of  the  valve  is  directly  under  the  eccentric  center  in  each  position. 

The  following  description  accompanies  Fig.  264: 

Position  I.  Admission. — Valve  displacement  =  S  =  steam  lap.  Steam  edge 
of  port  just  opening. 

Position  II.  Dead  Center. — Valve  displacement  =  S  +  I  =  steam  lap  +  lead. 
Valve  open  amount  of  lead  (Z). 

Position  III.  Maximum  Steam  Opening. — Valve  displacement  =  T/2  =  half  of 
valve  travel.  Valve  open  to  steam  T/2  —  S. 

Position  IV.  Cut-off. — Valve  displacement  =  S  =  steam  lap.  Steam  edge  of 
valve  just  closing. 

Position  V.  Valve  Central. — Valve  displacement  =  zero.  Steam  lap  S  and  ex- 
haust lap  E  are  measured  in  this  position. 

Position  VI.  Release. — Valve  displacement  =  E  =  exhaust  lap.  Exhaust  edge 
of  valve  just  opening. 


VALVES  AND  VALVE  GEARS 


413 


Position  VII.  Maximum  Exhaust  Opening. — Valve  displacement  =  T/2  =  half 
travel  of  valve.  Valve  open  to  exhaust  T/2  —  E. 

Position  VIIL  Compression. — Valve  displacement  =  E  =  exhaust  lap.  Ex- 
haust edge  of  valve  just  closing. 

From  the  valve  section  at  the  top  of  Fig.  264  the  following  equations 
are  derived: 


<£  Engine 


El.  Max.  Steam  Opening 


IP.  Cutoff 


yn.Max.Exhausf  Opening 
Y///////X  Y///////X 


YlH.  Compression 


FIG.  264. 


(usuaUy) 


(20) 
(21) 
(22) 


Formula  (22)  is  arbitrary  but  may  be  used  as  a  guide. 


414  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

For  minimum  value  of  W  (as  PI  >  P) : 


or: 


?~ 


E  -  B 


(23) 


Turning  the  indicator-  and  valve-section  diagrams  end  for  end,  and 
the  crank-eccentric  diagrams  through  an  angle  of  180  degrees  would  show 


l*L.  Engine    • 


the  crank-end  events,  but  this  is  unnecessary  when  the  angularity  of  the 
connecting  rod  is  neglected. 

When  familiarity  with  Fig.  264  is  attained  the  crank-eccentric  dia- 
gram may  be  used  alone  to  show  any  valve  setting. 

In  following  through  the  events,  valve  dimensions  were  assumed; 
in  practice  the  events  of  the  stroke  are  assumed,  or  it  may  be  said  that 
the  indicator  diagram  is  designed.  In  Fig.  265  a  crank-eccentric  diagram 
is  shown  for  the  head  end,  for  an  indicator  diagram  drawn  above  it. 
Any  crank  circle  may  be  taken,  angles  and  ratios  being  independent  of 
this.  A  valve  travel  may  be  assumed,  then  transferred  as  shown  to  any 


VALVES  AND  VALVE  GEARS  415 

other  value  by  graphical  proportion  when  one  dimension  (usually  Ps) 
is  known.  It  is  necessary  with  this  diagram  to  assume  a  lead  as  a  certain 
fraction  of  the  port  opening,  as  the  actual  valve  travel  may  be  different 
from  that  assumed. 

The  following  procedure  may  be  used :  Crank  positions  are  found  by 
projection  from  the  desired  indicator  diagram  as  shown.  The  angle 
between  crank  positions  for  admission  and  cut-off  is  obviously  equal  to 
the  angle  between  eccentric  positions  for  the  same  events.  The  projec- 
tion of  these  eccentric  positions  upon  the  horizontal  center  line  of  the 
valve  travel  circle  (the  eccentric  circle)  must  be  at  the  same  point,  as  the 
valve  must  be  in  the  same  position  for  both  events.  Then  connecting 
the  points  where  the  valve-travel  circle  cuts  the  crank  at  1  and  2,  and 
swinging  this  line  around  tangent  to  an  arc  of  radius  S  into  position  3-4 
parallel  to  the  vertical  center  line  as  shown,  the  eccentric  positions  for 
admission  and  cut-off,  the  steam  lap-angle  Q  and  the  steam  lap  S  are 
determined. 

The  exhaust  events  may  be  first  assumed,  the  line  5-6  being  swung 
around  to  position  7-8  tangent  to  the  arc  of  radius  E.  The  angular 
advance  may  thus  be  determined  by  admission  and  cut-off  (the  steam 
events),  or  by  release  and  compression  (the  exhaust  events).  If  deter- 
mined from  the  steam  events,  the  exhaust  lap  is  found  by  adjusting  a 
proper  relation  between  release  and  compression  for  the  angular  relation 
between  crank  and  eccentric  so  found.  If  determined  from  the  exhaust 
events  (which  is  unusual  except  for  engines  with  a  releasing  gear),  the 
steam  lap  would  be  found  from  the  best  relation  of  admission  to  cut-off 
possible  with  this  angular  relation.  Sometimes,  especially  in  locomotives, 
exhaust  lap  is  zero  or  may  even  be  negative,  when  the  valve  is  open  to 
exhaust  at  both  ends  when  in  the  central  position;  it  is  then  known  as 
exhaust  clearance. 

Effect  of  Rod  Angularity. — In  Fig.  266- A  for  the  Scotch  yoke,  the 
horizontal  distance  between  crank  pin  1  and  crosshead  pin  2  is  the  same 
for  all  positions  of  the  crank  pin.  Let  this  distance  L  be  the  length  of 
the  connecting  rod  of  a  slider-crank  mechanism;  then  for  the  same 
angular  movement  of  the  crank  from  the  head-end  dead  center,  the  cross- 
head  pin  of  Fig.  266-A  moves  the  distance  a,  while  that  of  Fig.  266-B 
moves  the  distance  6.  This  is  due  to  the  angularity  of  the  rod  and  it  is 
obvious  that  the  crosshead  and  piston  of  Fig.  266-5  will  always  be  nearer 
to  the  crank  than  that  of  Fig.  266-A  except  in  the  dead-center  positions. 

In  valve  diagrams  it  is  convenient  to  consider  piston  displacement  in 
connection  with  the  crank  circle.  Then  in  Fig.  266-5,  with  the  center 
of  the  crosshead  pin  as  a  center,  draw  an  arc  of  radius  L  cutting  the  center 


416 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


line  of  engine;  this  marks  off  the  distance  6  which  is  the  piston  displace- 
ment from  the  head  end  of  the  stroke.  Thus  for  any  position  of  the  crank 
pin  the  corresponding  piston  position  may  be  shown  on  the  crank  pin 
circle  diameter.  The  same  is  true  for  valve  displacement  and  the 
eccentric  circle,  but  the  maximum  angularity  of  the  eccentric  rod  is 
usually  so  small  that  it  is  neglected. 

It  is  usual  to  neglect  angularity  in  drawing  valve  diagrams,  at  least 
for  preliminary  work;  the  events  used  are  then  called  nominal  cut-off, 
compression,  etc.  To  find  the  actual  events,  the  method  just  given  is 
used  and  this  is  shown  in  Fig.  267.  It  will  be  noticed  that  all  events  of 


FIG.  266. 

the  stroke  from  head  to  crank  end  occur  later  in  the  stroke  than  the 
nominal  events,  while  those  from  crank  to  head  end  occur  earlier. 

Actual  indicator  diagrams  are  as  shown  in  Fig.  267,  if,  with  the  piston 
at  the  head  end  of  the  stroke,  the  pencil  of  the  indicator  is  to  the  right  of 
the  paper  clips  as  at  A,  Fig.  268,  the  drum  moving  as  indicated  by  the 
arrow  as  the  cord  is  unwound;  if  for  this  piston  position  the  arrangement 
is  as  at  B,  Fig.  268,  the  indicator  diagrams  will  be  changed  end  for  end. 

The  diagram  of  Fig.  265  has  been  given  because  it  shows  the  actual 
relation  of  crank  and  eccentric,  and  the  principles  of  valve  motion  may 
more  clearly  be  seen  than  with  any  other  diagram.  Moreover,  it  is 
general  in  its  application  and  may  be  used  for  any  eccentric-driven  gear 
in  conjunction  with  other  diagrams  showing  the  motion  of  wrist  plate, 
linkage  or  cams.  For  the  single  valve  it  has  limitations  except  as  the 


VALVES  AND  VALVE  GEARS 


417 


trial-and-error  method  is  used;  this  is  also  true  of  the  Reuleaux  and 
Zeuner  diagrams.  Therefore  no  further  use  will  be  made  of  this  diagram 
for  the  single  valve  except  to  explain  principles,  but  a  diagram  having 


k__ 

1 

Head  End  Diagrams 


Crank  End  Diagrams 


FIG.  267. 


all  the  advantages  and  none  of  the  disadvantages  of  the  other  diagrams 
for  the  single  valve  will  now  be  explained. 

The  Bilgram  Diagram. — Starting  with  the  crank  at  the  head-end  dead 
center  and  with  angular  advance  A,  Fig.  269,  move  crank  and  eccentric 
through  angle  6.  Distance  1-2,  perpendicular  to  vertical  center  Line, 
represents  valve  displacement  from  central  position  by  the  method  of 
Fig.  265.  Produce  the  new  crank  position  as  shown,  forming  angle  0 
below  the  horizontal  center  line  and  to  the  right  of  the  vertical  center 
line.  Now  lay  off  angle  A  (the  angular  advance)  above  the  horizontal 
center  line  as  shown  by  the  heavy  line.  The 
intersection  of  this  line  with  the  valve-travel  circle 
is  the  eccentric  center  of  the  Bilgram  diagram, 
being  a  fixed  point  for  any  given  valve,  setting. 
Drop  the  perpendicular  3-4  to  the  new  position  of 
the  crank  produced.  Then  triangles  1-2-0  and  3-4-0  are  equal,  as  each 
contains  equal  sides  1-0  and  3-0  (being  radii  of  the  same  circle),  one  right 
angle,  and  the  angle  A  +  9.  Then  line  3-4  equals  line  1-2,  the  dis- 
placement of  the  valve  from  the  central  position.  This  is  true  for  all 

27 


FIG.  268. 


418 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


positions  of  the  crank,  the  perpendicular  3-4  sometimes  falling  on  the 
center  line  of  the  crank  and  sometimes  on  this  line  produced  beyond  the 
center  of  the  circle. 


.C.L.  Engine    V 


I 


Crank  Circle  =  Stroke  of  Piston - 
FIG.  269 


FIG.  270.— The  Bilgram  diagram. 

For  convenience  of  application  the  lap  circles  are  drawn  about  the 
eccentric  center  of  the  Bilgram  diagram,  the  radius  of  the  circle  equalling 
the  lap.  This  is  shown  in  Fig.  270,  drawn  with  the  same  data  as  Fig. 


VALVES  AND  VALVE  GEARS  419 

264,  with  which  it  may  be  compared.     With  the  same  numbers  for  crank 
positions  the  following  description  will  make  the  diagram  clear: 

I.  Admission. — Valve   displacement    (1-2)  =  S  =  steam  lap.     Steam  edge  of 
valve  just  about  to  open. 

II.  Dead     Center. — Valve      displacement       (1-3)  =  S  + 1  =  steam  lap  +  lead. 
Valve  open  amount  of  lead  I. 

III.  Maximum  Steam  Opening. — Valve  displacement  (1-0)  =  T/2  —  half  of  valve 
travel.     Valve  open  to  steam  (0-4)  =  Ps  =  T/2  -  S. 

IV.  Cut-off. — Valve    displacement    (1-5)  =  S  —  steam    lap.    Steam    edge    of 
valve  just  closing. 

V.  Valve  Central. — Valve  displacement  =  zero. 

VI.  Release. — Valve  displacement  (1-6)  =  E  =  exhaust  lap.    Exhaust  edge  of 
valve  just  opening. 

VII.  Maximum  Exhaust  Opening. — Valve  displacement  (1—0)  =  T/2  =  half  travel 
of  valve.     Valve  open  to  exhaust  (0-7)  =  PE  =  T/2  —  E. 

VIII.  Compression. — Valve    displacement    (1-8)  =  E  =  exhaust    lap.     Exhaust 
edge  of  valve  just  closing. 

Fig.  270  is  the  head-end  diagram;  for  the  crank-end  diagram  the 
location  of  the  lap  circles  and  the  crank  positions  for  the  various  events 
are  diametrically  opposite. 

In  applying  the  diagram  the  cut-off,  maximum  steam  port  opening 
Ps  and  lead  I  are  usually  assumed.  The  cut-off  position  of  the  crank, 
lead  line  and  arc  of  radius  Ps  are  drawn  as  shown;  then  the  steam  lap 
circle  is  drawn  so  as  to  come  tangent  to  these  three  lines,  fixing  the  center 
of  eccentric,  angular  advance  and  valve  travel.  The  compression  is  the 
exhaust  event  usually  assumed;  producing  this  line  as  shown  dotted, 
the  exhaust  lap  circle  is  drawn  tangent  to  it  from  the  eccentric  center, 
then  the  release  position  is  tangent  to  the  other  side.  Should  release 
and  compression  exchange  positions,  release  would  occur  before  com- 
pression (in  fraction  of  stroke);  exhaust  lap  would  then  be  negative 
(the  valve  would  be  open  to  exhaust  at  both  ends  when  valve  is  at  center 
of  travel,  giving  exhaust  clearance)  and  the  circle  would  be  a  dotted 
line.  The  effect  of  angularity  of  connecting  rod  is  found  as  in  Fig.  267. 

Double-ported  Valves. — There  are  various  designs  of  multi-ported  valves, 
but  the  principle  of  all  is  to  increase  the  valve  opening  with  a  given 
travel.  The  principle  of  operation  of  a  double-ported,  balanced  slide 
valve  is  shown  in  Fig.  271.  An  adjustable  balance  plate  (or  pressure 
plate)  has  depressions  to  match  the  ports,  the  pressure  being  kept  from 
all  portions  of  the  valve  not  opposite  the  ports  by  the  valve  seat  and 
balance  plate.  At  A  the  valve  is  displaced  to  the  right  by  the  amount  of 
the  steam  lap  and  is  just  about  to  open  the  head-end  port.  At  B  it 
has  moved  from  the  edge  of  the  port  a  distance  equal  to  the  width  of  the 


420 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


auxiliary  port  through  the  valve;  up  to  this  time  the  opening  has  been 
double  that  indicated  by  the  movement,  as  steam  enters  at  two  edges, 
but  any  further  movement  up  to  the  position  shown  at  C  will  not  increase 
the  opening,  as  the  port  through  the  valve  is  closed  as  fast  as  the  cylinder 
port  is  opened.  Sometimes  the  port  opening  equals  the  width  of  the 
auxiliary  port  before  the  valve  reaches  the  position  B  (where  the  auxiliary 
port  begins  to  close) ;  then  the  opening  is  as  for  a  single  port  till  position 
B  is  reached  and  constant  port  opening  begins.  The  valve  is  at  the 


FIG.  271. — Double-ported  slide  valve. 

extreme  left  of  its  travel  at  D,  uncovering  the  exhaust  port;  a  double- 
ported  effect  at  the  exhaust  edge  is  not  necessary  with  a  single  valve  as 
the  exhaust  lap  is  small  and  there  is  usually  a  wide  opening. 

Some  double-ported  valves  give  a  double  opening  throughout  their 
entire  travel,  as  the  Corliss  valve  of  Fig.  259. 

Shifting  and  Swinging  Eccentric. — In  Chap.  XIX,  illustrations  are  given 
of  governors  in  which  the  angular  position  and  radius  of  the  eccentric  are 
changed  when  the  load  changes.  If  the  eccentric  moves  in  a  straight 
line  it  is  called  a  shifting  eccentric,  and  if  it  swings  from  a  pivot  in  the 
arc  of  a  circle,  a  swinging  eccentric, 


VALVES  AND  VALVE  GEARS 


421 


The  diagram  of  Fig.  265  will  first  be  used  to  show  the  changes,  which 
are  given  for  different  methods  in  Fig.  272.  At  A  is  shown  an  older  style 
in  which  the  eccentric  is  rotated  about  the  shaft.  The  angular  advance 
is  increased  with  an  increasing  lead,  but  the  valve  travel  is  not  changed. 
Full  lines  show  the  setting  for  maximum  cut-off  and  dotted  lines  for  a 
shorter  cut-off. 


A  shifting  eccentric  is  shown  at  B  in  which  both  valve  travel  and  angu- 
lar advance  are  changed,  as  is  also  the  case  with  the  swinging  eccentric 
shown  at  C.  The  pivot  for  Fig.  272-C  was  located  at  the  crank  side -of 
the  vertical  center  line,  but  it  is  sometimes  at  the  right.  The  pivot  was 
located  to  give  zero  lead  when  the  angular  advance  is  90  degrees.  The 
lead  is  constant  for  B. 


These  diagrams  are  not  convenient  for  the  actual  design  of  the 
changing  eccentrics,  so  the  same  three  cases  are  shown  for  the  Bilgram 
diagram  in  Fig.  273.  Crank  positions  for  all  events  of  the  stroke  (for 
the  head  end)  are  shown  by  the  full  lines,  and  for  the  shortened  cut-off 
by  the  dotted  lines;  the  decreased  valve- travel  circle  for  the  short 
cut-off  is  omitted  as  it  serves  no  useful  purpose.  The  path  of  the  eccen- 


422  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

trie  change  is  drawn  as  though  the  crank  pin  were  at  the  bottom  of  the 
circle  and  rotates  counterclockwise. 

It  will  be  noticed  that  the  maximum  steam  port  opening  Ps  (see  Fig. 
265)  has  decreased  considerably  for  the  shorter  cut-off  of  B  and  C;  it 
may  easily  be  shown  that  for  one-quarter  cut-off  the  opening  would 
be  extremely  small  unless  the  valve  travel  and  steam  lap  were  very  large. 
It  is  for  this  reason  that  double-  and  triple-ported  valves  are  used,  giving 
double  or  treble  the  opening  indicated  by  the  Bilgram  diagram,  which 
gives  the  nominal  port  opening. 

Rectangular  valve  diagrams  are  diagrams  in  which  ordinates  represent 
valve  displacement,  and  abscissas  either  piston  or  crank  displacement; 
if  the  latter,  the  diagram  is  plotted  on  a  rectified  crank  circle.  As  dis- 
placement is  the  sum  of  lap  and  port  opening,  such  diagrams  may  be  used 
to  show  port  opening  during  the  cycle,  and  may  show  the  point  on  piston 
path  or  crank  circle  at  which  the  valve  event  occurs. 

To  illustrate  a  rectangular  diagram  on  the  piston  path,  together  with  a 
double-ported  valve,  shifting  eccentric  and  some  of  the  principles  of 
Par.  142,  the  design  of  a  high-speed  engine  will  be  assumed  with  data  as 
follows:  Cylinder  diameter  12  in.,  stroke  18  in.  and  r.p.m.  200;  initial 
gage  steam  pressure  125  lb.,  back  pressure  15  Ib.  absolute  and  clearance 
6  per  cent. 

The  nominal  steam  velocities  from  (3)  and  (4)  are:  Vs  =  7250  and 
VE  =  5180.  The  ports  in  the  cylinder  must  be  determined  from  VE', 

then  (2)  gives: 

113  X  600       1Q  t 
aE=    -5180-     =13.1sq.m. 

Assuming  the  length  of  the  port  equal  to  the  cylinder  diameter,  the  width 
is: 

.10,          =  1.09,  say 


The  maximum  steam  port  opening  will  be  determined  from  Vs,  which 
from  (2)  gives: 

113  X  600 

as=     -7250-     =9'38 
and 

Ps=^=0.78.       '    -,  - 

On  account  of  the  double-ported  valve  and  the  desirability  of  having 
sufficient  port  opening  at  short  cut-offs,  assume  the  nominal  port  opening 
which  will  be  used  for  the  Bilgram  diagram  at  maximum  cut-off  to  be 


VALVES  AND  VALVE  GEARS 


423 


equal  to  WE  or  1%  in.     Also  assume  the  maximum  cut-off  to  be  %  stroke 
and  the  nominal  lead  (that  used  on  the  Bilgram  diagram)  to  be  constant 


and  equal  to  H 6  in.     We  are  now  ready  to  start  the  Bilgram  diagram  and 
this  is  shown  in  Fig.  274  at  the  top. 


424  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

We  must  assume  compression  or  release  to  find  the  exhaust  lap. 
This  is  rather  arbitrary  and  is  done  in  various  ways.  It  is  sometimes 
determined  by  assuming  that  nominal  compression  raises  the  compression 
pressure  to  initial  pressure  when  the  cut-off  is  very  short.  With  the 
clearance  and  compression  assumed,  initial  pressure  is  reached  with  a 
nominal  compression  of  0.56  stroke  (see  Chap.  XII).  Let  us  assume  that 
with  0.1  cut-off,  nominal  compression  is  0.5  stroke.  Laying  this  off  on 
the  Bilgram  diagram  the  exhaust  lap  is  found  as  shown. 

The  following  data  are  obtained  from  the  diagram:  Valve  travel, 
4%  in.;  steam  lap,  iKe  in.;  exhaust  lap,  %2  m- 

The  displacement  curve  on  the  piston  path  is  called  the  valve  ellipse; 
this  is  sometimes  drawn  by  neglecting  the  angularity  of  the  connecting 
rod,  but  this  is  taken  into  account  in  Fig.  274,  the  rod  being  5  cranks  long. 
The  crank  circle  is  usually  divided  into  a  number  of  equal  parts  (they 
need  not  be  equal  but  it  is  usually  more  convenient)  from  which  the  crank 
positions  are  drawn;  the  correct  piston  position  is  found  by  drawing  the 
arc  to  the  center  line  as  shown  for  the  crank  position  marked  Ko  cut-off, 
and  this  is  produced  to  the  center  of  the  valve  ellipse  (not  a  true  ellipse) 
from  which  valve  displacements  are  measured.  The  lap  lines  are  then 
drawn  in,  the  subscripts  H  and  C  denoting  head  and  crank  end;  should 
exhaust  lap  be  negative  it  is  drawn  on  the  same  side  of  the  center  as  the 
steam  lap,  as  it  is  on  the  same  side  of  its  respective  port  edge.  Ordinates 
measured  from  these  lap  lines  to  the  ellipse  give  valve  openings  for  a 
single-ported  valve  at  any  point;  it  is  then  easy  to  find  the  part  of  the 
stroke  at  which  the  valve  is  opened  and  closed.  The  ellipse  is  tangent  at 
the  ends  at  the  ordinate  SH  -f  I-  The  upper  valve  ellipse  is  for  %  cut-off 
(the  maximum)  and  the  lower  for  y±  cut-off.  In  both  ellipses  head-end 
cut-off  occurs  at  6,  head-end  release  at  d  and  head-end  compression  at  e] 
the  subscript  1  denotes  the  same  for  the  crank  end.  Admission  is  so 
near  the  ends  of  the  stroke  that  it  may  not  be  well  shown  in  this  diagram 
unless  drawn  to  a  large  scale. 

The  diagram  may  be  easily  modified  for  the  double-ported  valve 
without  changing  in  any  way  the  timing  of  the  events;  the  only  difference 
is  the  increased  steam  valve  opening.  The  valve  is  shown  to  scale  in 
Fig.  271. 

There  are  different  methods  of  proportioning  the  auxiliary  port  PA 
through  the  valve;  in  this  case  it  has  been  taken  as  one-half  of  Ps.  The 
port  in  the  cylinder  at  the  valve  face  must  be  equal  to  Bv  +  2PA  (=  Bv 
+  PS)  as  shown  in  Fig.  271. 

To  show  the  effect  of  the  double  port  draw  lines  on  the  ellipses  a  dis- 
tance PA  from  the  steam  lap  lines.  During  this  displacement  the  opening 


VALVES  AND  VALVE  GEARS 


425 


is  twice  that  due  to  the  displacement,  the  piston  travelling  up  to  the 
point  where  the  line  crosses  the  ellipse;  after  this  the  opening  is  constant 
as  explained  in  connection  with  Fig.  271.  When  the  ellipse  again  crosses 
this  line  the  valve  begins  to  close  again,  closing  at  b  as  before  the  double 
port  was  added.  The  opening  diagram  now  has  the  form  afgb.  For 
the  one-quarter  cut-off  Ps  is  less  than  PA,  so  that  the  double  opening 
obtains  all  the  time  the  valve  is  open  as  shown  at  h. 

Piston-velocity  curves  drawn  with  a  crank-circle  radius  equal  to 
Ps  for  the  %  cut-off  are  shown  plotted  on  the  steam  and  exhaust  lap 
lines  for  the  long  cut-off  in  dotted  lines,  and  for  the  short  cut-off  on  the 
steam-lap  line  as  far  as  the  valve  is  open.  It  will  be  seen  that  in  all 
cases  the  valve-opening  curve  rises  quicker  than  the  velocity  curve  at 


0          I  3          4-          5         6          7          8          9         10         //        12 


first,  and  for  the  long  cut-off  lies  above  till  maximum  steam  port  opening 
has  been  reached.  The  exhaust  opening  curve  is  above  it  for  three- 
quarters  of  the  stroke.  For  the  short  cut-off,  the  velocity  line  is  crossed 
early  by  the  valve-opening  line.  The  effect  upon  the  steam  pressure 
depends  upon  the  value  of  nominal  steam  velocity  V  assumed  in  deter- 
mining the  ports,  but  it  is  obvious  that  for  any  type  of  gear  in  which  the 
valve  closes  gradually,  the  ratio  of  port  opening  to  piston  velocity  de- 
creases very  rapidly  beyond  a  certain  point,  and  the  relative  steam  velo- 
city increases;  if  this  velocity  is  very  high  the  pressure  head  increases 
and  wire-drawing  occurs — a  very  common  experience.  Gradual  closing 
of  the  exhaust  port  gives  the  effect  of  an  early  compression  and  is  not 
objectionable,  as  allowance  may  be  made  for  it. 

The  method  of  plotting  a  piston  displacement  curve  on  a  rectified 
crank  circle,  accounting  for  angularity  of  the  connecting  rod,  is  shown 
in  Fig.  92,  Chap.  XIII  This  forms  part  of  a  rectangular  valve  diagram 
which  is  shown  in  Fig.  275.  The  valve  displacement  curve  may  be  plotted 
in  the  same  way,  although  the  angularity  of  the  eccentric  rod  is  usually 


426  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

neglected.  By  placing  this  curve  in  correct  phase  relation  with  the  piston 
displacement  curve  the  crank  position  for  any  given  event  may  be  deter- 
mined, from  whence  the  piston  position  may  be  found.  This  is  traced 
through  for  head-end  release  by  dotted  lines,  the  letters  denoting  the 
same  as  in  Fig.  274.  Fig.  275  is  for  the  single-ported  valve  with  %  cut- 
off in  Fig.  274. 

This  diagram  is  very  valuable  for  complicated  gears  difficult  to  handle 
in  any  other  way.  One  or  more  valve  curves  may  be  used,  drawn  on 
separate  pieces  of  tracing  cloth  so  that  their  relative  positions  may  be 
changed.  It  is  sometimes  convenient  to  draw  the  lap  lines  parallel  to 
the  curves,  especially  when  an  auxiliary  valve  is  used.  This  diagram  will 
be  referred  to  in  connection  with  sleeve  valves  for  internal-combustion 
engines. 

Valve  Stems. — Let  d  be  the  diameter  of  the  stem,  a  the  unbalanced 
area  of  the  valve,  fj,  the  coefficient  of  friction,  p  the  unbalanced  steam 
presures  in  Ib.  per  sq.  in.  and  S  the  stress  in  the  stem.  Then: 

-j—  S  =  fj,pa 

or: 

d  =  J^  (24) 

\      7TO 

For  practical  results,  a  high  coefficient  of  friction  and  low  stress  must 
be  used.  Assume  M  =  0.25,  S  =  5000  and  p  =  125. 

145.  Corliss  Releasing  Gear. — It  is  probable  that,  strictly  speaking, 
the  Corliss  gear  is  a  releasing  gear  with  rocking  valves  and  a  wrist  plate 
operated  by  a  single  eccentric;  however,  the  name  has  been  applied  to 
engines  with  two  eccentrics  and  a  double  wrist  plate,  and  even  when 
a  steam  wrist  plate  is  omitted.  It  is  also  applied  to  nonreleasing  gears 
with  shaft  governor  and  swinging  eccentric,  although  this  usage  is  opposed 
by  some  engineers. 

In  Fig.  276-A,  a  valve  is  shown  at  the  extreme  left  of  its  travel, 
fully  opening  the  port  at  the  right;  as  it  uncovers  the  port  at  the  left  it 
travels  the  distance  T.  At  B,  the  central  arm  of  a  double  bell  crank 
travels  the  same  horizontal  distance.  It  is  so  connected  to  the  two  valves 
that  it  begins  to  open  the  port  at  the  same  point  in  its  horizontal  projec- 
tion that  A  does,  and  opens  the  valve  the  same  width  at  the  end  of  its 
travel.  The  distance  traveled  by  the  valve  is  T\,  much  less  than  T. 
The  valve  starts  to  open  slowly  due  to  the  dead-center  effect  of  the  bell 
crank  arm,  but  opens  quickly  when  the  port  edge  is  reached.  This  is 
the  effect  of  the  wrist  plate.  If  another  bell  crank  is  used  as  at  C,  and 


VALVES  AND  VALVE  GEARS  427 

the  movement  is  made  from  the  full  to  dotted  position,  this  effect  is  still 
more  marked  due  to  the  more  rapid  increase  of  angle  a  as  the  valve  moves 
to  the  right.  This  effect  is  also  obtained  in  angle  5  of  the  main  bell 
crank.  The  relative  positions  of  the  bell  cranks  and  the  change  of  angu- 
larity of  the  rod  connecting  them  has  influence,  and  advantage  is  taken 
of  these  principles  in  different  ways  in  Coiliss  and  other  gears. 

In  designing  the  Corliss  gear  two  diagrams  are  employed,  one  for  the 
wrist  plate  and  valve  levers  and  another  for  the  crank  and  eccentric. 
The  latter  is  sometimes  omitted,  the  laps  being  taken  from  a  table,  the 
values  of  which  are  usually  for  the  old,  slow-speed  type  with  very  late 
compression  and  are  not  generally  applicable.  Some  idea  of  these 
diagrams  is  given  in  Chap.  III. 


7M         W7//b7777777? 
i-^-H 


FIG.  276. 

The  crank-eccentric  diagram  is  that  of  Fig.  265.  The  cut-off  is 
effected  by  the  governor  and  trip  mechanism,  so  the  cut-off  may  be  neg- 
lected in  the  diagram;  it  may  be  found  as  in  Fig.  265,  neglecting  lead, 
and  gives  the  maximum  cut-off  should  the  load  be  too  great  for  the  engine, 
and  the  governor  fail  to  trip  the  gear.  The  distances  S  and  E  will  be 
used  for  convenience  but  are  not  the  laps;  starting  with  the  wrist  plate 
at  the  center  of  its  travel,  they  give  the  positions  of  the  eccentrics  which 
will  move  the  valves  by  the  amount  of  their  laps  (lap  plus  lead  in  the 
case  of  the  steam  valves). 

In  most  diagrams  given  in  text-books,  very  late  compression  is  as- 
sumed, giving  a  small  angular  advance;  the  resulting  maximum  cut-off 
under  governor  control  is  therefore  about  0.375  to  0.4  stroke  and  presents 
no  difficulties  for  reasonable  overloads.  In  laying  out  the  diagram 


428 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


in  this  paragraph,  the  indicator  diagram  of  the  Corliss  example  (20  X 
48  —  100)  of  Chap.  XII  is  used.  The  problem  calls  for  a  long-range  cut- 
off, but  the  single-eccentric  layout  will  be  first  taken  up.  This  shows  the 
limit  of  capacity  when  a  rather  early  compression  (0.8  stroke)  is  used,  and 
shows  why  so  late  a  compression  and  release  as  are  practicable  are 
generally  used;  also,  in  conjunction  with  the  diagrams  of  Chap.  XIII, 
why  double  eccentrics  with  separate  steam  and  exhaust  wrist  plates  were 


FIG.  277. 

Used  on  the  low-pressure  cylinders  of  compound  Corliss  engines  even 
though  the  high-pressure  cut-off  gave  ample  overload  capacity  with  a 
single  wrist  plate.  The  single-wrist-plate  diagram  is  shown  in  Fig.  277 
and  the  crank-eccentric  diagram  in  Fig.  278.  These  are  for  the  head  end, 
and  as  the  gear  is  so  adjustable  and  cut-off  depends  upon  governor  con- 
nections, angularity  of  the  connecting  rod  is  neglected  in  this  paragraph. 
Procedure. — The  diagram  for  wrist  plate  and  eccentric  may  be  first 
drawn.  Until  considerable  experience  is  had,  this  may  require  a  number 
of  trials,  but  for  a  giyen  compression  and  release,  formulas  may  be  made 


VALVES  AND  VALVE  GEARS 


429 


which  greatly  reduce  labor.  The  method  by  which  Figs.  277  and  278 
were  drawn  has  been  used  by  the  author  for  several  years  with  satisfac- 
tion and  dispatch.  The  dimensions  given  are  those  practically  required 


FIG.  278. 


with  the  exception  of  the  angles,  which  are  omitted.  For  standard 
gears  already  designed,  and  usually  intended  for  several  cylinder  dia- 
meters, rs  and  rE  are  fixed.  For  new  work  their  value  is  more  or  less 


430  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

arbitrary,  the  minimum  depending  upon  the  detail  design  of  the  mechan- 
ism. In  Fig.  277  they  are  taken  equal  to  d,  the  valve  diameter,  which 
in  this  case  was  computed  by  the  formulas  of  Par.  142.  Rs  was  taken 
asr,s  +  1.5  in.  and  RE  =  l.IRs-  The  radius  RT  of  the  main  wrist- 
plate  pin  depends  upon  the  design  of  the  reach-rod  releasing  mech- 
anism, but  its  horizontal  travel  is  T.  Draw  arcs  of  radii  rs  and  rE  from 
valve  centers;  from  center  of  wrist  plate  draw  arcs  of  radii  Rs,  RE  and 
RT.  Draw  lines  from  center  of  wrist  plate  tangent  to  arcs  of  rs  and  rE. 
For  new  work  draw  valve  levers  normal  to  these  lines.  For  old  designs 
the  levers  must  be  so  located  that  the  travel  of  the  dashpot  pin  at  radius 
rA  will  be  equally  divided  by  the  horizontal  center  line,  but  the  normal 
line  may  be  used  tentatively.  Now  locate  a  point  on  arc  Rs  below  the 
tangent  line,  and  on  arc  RE  above  the  tangent  line.  This  is  arbitrary, 
but  is  made  about  11.5  degrees  in  Fig.  277.  Old  practice  made  this  point 
on  the  line  and  for  very  late  compression  this  is  permissible;  but  the 
method  shown  reduces  the  travel  when  the  valve  is  closed.  It  necessi- 
tates a  greater  angle  of  wrist-plate  motion  0,  which  is  69  degrees  in  this 
design. 

Now  connect  this  point  with  the  valve  lever  position  already  found; 
locate  other  extreme  wrist-plate  positions  by  angle  0,  which  is  arbitrary 
and  found  by  trial;  and  bisect,  giving  the  position  when  the  wrist  plate  is 
at  the  center  of  its  travel.  With  the  length  of  connecting  link  found  for 
the  first  extreme  position  laid  off  with  a  beam  compass,  find  the  valve 
lever  positions  to  correspond  with  the  other  wrist-plate  positions. 

For  single-ported  valves  the  dimensions  would  be  different,  but  these 
are  not  much  used. 

Draw  in  one  of  the  ports  in  any  position,  the  location  of  the  second 
port  being  found  later.  Show  the  steam  valve  edge  dotted  in  the  lead 
position  which  may  be  taken  from  KG  to  J^  of  the  port  width;  the  latter 
was  assumed. 

Now  draw  the  crank-eccentric  diagram  for  the  compression  and 
release  desired;  the  latter  is  taken  arbitrarily  and  not  far  above  the 
center  line  as  shown;  in  Fig.  278  it  is  ]/±  in.  on  a  5  in.  circle.  The  angular 
advance  may  be  found  as  described  for  Fig.  265.  Now  draw  the  crank 
on  the  head-end  dead  center.  The  eccentric  circle  may  be  made  equal 
to  T  if  this  is  greater  than  TE\  its  value  is  unimportant.  For  the  steam 
valve,  cut  the  angular  advance  position  with  arc  of  radius  Ts/2;  this 
gives  the  virtual  eccentric  radius  to  give  the  steam  wrist-plate  arm  the 
travel  Ts.  Lay  off  S  on  Fig.  277,  giving  the  position  of  the  wrist-plate 
pin  when  the  crank  is  on  the  dead  center.  Find  the  corresponding  valve 
lever  position;  this  is  shown  dotted.  This  position  corresponds  with 


VALVES  AND  VALVE  GEARS  431 

the  lead  position  of  the  valve;  taking  this  distance  along  the  valve  circle 
on  a  divider,  the  three  positions  (two  extreme  and  central)  of  the  valve 
may  be  found.  The  position  corresponding  to  the  central  position  of  the 
wrist  plate  gives  the  steam  lap. 

It  will  be  noticed  that  when  the  valve  is  wide  open  there  is  some  over- 
travel;  this  gives  a  little  quicker  valve  opening. 

At  B  draw  the  valve  edge  in  its  extreme  position;  this  is  some  greater 
than  that  indicated  at  A  due  to  the  wrist-plate  arm  crossing  its  dead 
center,  and  besides  this  some  clearance  must  be  allowed  the  latch  hook; 
but  this  is  usually  small  and  may  be  overlooked  if  ample  allowances  are 
made.  When  in  position  B,  the  back  edge  of  the  valve  must  have  suffi- 
cient seal  to  prevent  leakage;  this  was  made  %  in.  in  Fig.  277,  but  may 
be  greater  for  large  engines  and  less  for  smaller  engines.  This  determines 
one  valve  face.  Now  move  this  face  to  its  extreme  open  position  as  at  C. 
It  is  well  to  allow  a  small  amount  for  adjustment,  and  discrepancies  in 
the  cylinder  casting  as  shown.  This  determines  the  bridge  width.  Draw 
a  center  line  through  the  bridge  as  shown  and  locate  on  the  vertical  center 
line  of  the  valve  chamber  as  at  D.  The  valve  design  may  be  now 
completed. 

The  exhaust  valve  may  be  treated  in  the  same  way,  the  dotted  posi- 
tion corresponding  to  compression,  when  the  valve  is  "  line-and-line " 
(just  closing).  Dimensions  may  be  determined  by  BI,  C\  and  DI  as 
for  the  steam  valve.  The  exhaust  valve  is  sometimes  made  symmetrical 
as  with  the  steam  valve. 

The  dotted  lines  across  the  valves  in  D  and  DI  show  the  slot  for  the 
tee-head  of  the  valve  stem;  it  is  located  so  that  it  is  vertical  when  the 
valve  edge  is  halfway  between  line-and-line  and  full  closed  positions, 
thus  allowing  for  wear. 

Valve-opening  diagrams  for  steam  and  exhaust  valves  are  shown  above 
and  below  the  crank-eccentric  diagram  respectively  in  Fig.  278.  The 
steam  valves  open  quickly,  the  more  so  because  of  the  over-travel. 
A  piston  velocity  curve  is  shown  dotted,  but  the  opening  curve  is  above 
it  until  past  the  point  of  maximum  cut-off.  The  theoretical  point  of 
maximum  cut-off  is  found  by  placing  the  eccentric  to  the  extreme  right 
of  its  travel;  this  places  the  wrist  plate  in  its  extreme  position,  and  if 
cut-off  does  not  occur  before  this  time  the  valve  is  not  unhooked  and  the 
opening  curve  is  that  shown  by  the  full  line.  But  past  the  point  when 
the  eccentric  and  wrist  plate  are  at  the  extreme  right  the  governor  does 
not  control,  so  that  this  gives  the  limit  of  valve  release.  However,  it 
takes  some  appreciable  time  for  the  dashpot  to  close  the  valve,  so  that 
the  real  cut-off  is  later. 


432  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

For  the  early  compression  assumed,  the  angular  advance  is  so  great 
that  the  maximum  cut-off  under  governor  control  is  but  Y±  stroke;  this 
is  at  b,  the  valve  being  released  at  this  point.  Furman  says  that  a  good 
dashpot  will  close  the  valve  in  Jfe  sec->  which  for  100  r.p.m.  would  be 
about  0.1  of  a  revolution.  Assuming  the  valve  to  be  entirely  closed  in 
this  time,  the  piston  would  travel  the  horizontal  distance  between  b 
and  c;  further  assuming  the  closing  curve  between  these  two  points  to  be 
a  straight  line,  the  actual  displacement  curve  would  be  abcic  and  the 
opening  of  the  port  would  follow  the  line  afghjh.  It  will  be  noticed  that 
the  port  is  wide  open  at  0. 1  stroke.  The  actual  closing  for  maximum  cut- 
off begins  at  about  0.29  and  ends  at  0.43  stroke;  a  relatively  slower  dash- 
pot  would  give  a  later  cut-off,  and  if  the  speed  were  high,  would  show 
signs  of  wire-drawing. 

In  Fig.  277,  the  exhaust  valve  shows  no  over-travel  but  the  opening  is 
quick;  it  is  not  likely  that  the  opening  would  be  less  than  shown  as  the 
compression  is  not  apt  to  be  earlier  than  assumed,  but  if  some  over-travel 
is  desired,  RE  may  be  increased,  thus  increasing  the  angle  of  valve  travel. 
The  displacement  curve  needs  no  modification  as  there  is  no  releasing 
mechanism  for  the  exhaust  valve. 

Lang-range  Cut-off. — These  gears  were  formerly  furnished  with  two 
eccentrics  and  a  double  wrist  plate,  the  steam  and  exhaust  portions 
working  independently.  For  the  design  considered  the  exhaust  wrist 
plate  may  be  exactly  as  determined,  so  will  not  be  redrawn.  A  wrist 
plate  is  sometimes  used  for  the  steam  valves,  but  the  travel  of  the  valve 
from  the  central  wrist-plate  position  to  full  closed  position  is  nearly  the 
same  as  to  the  full  open  position,  therefore,  the  wrist-plate  pin  must  be 
located  nearer  the  top  and  the  angular  motion  much  reduced.  This  led 
to  abandoning  the  wrist  plate  for  the  steam  valve  and  this  will  be  as- 
sumed; then  in  Fig.  279  the  wrist  plate  is  not  shown.  Fig.  280  is  the 
crank-eccentric  diagram. 

As  in  this  problem  a  %  cut-off  under  governor  control  is  desired,  the 
crank  must  be  in  this  position  as  the  wrist  plate  and  eccentric  are  at  the 
extreme  right;  moving  the  crank  back  to  head-end  dead  center  brings  the 
eccentric  back  of  the  vertical  center,  giving  negative  angular  advance. 
As  the  valve  is  open  the  amount  of  the  lead  in  this  position,  it  is  obvious 
that  when  the  wrist  plate  is  at  the  center  of  its  travel  the  valve  is  open, 
giving  negative  steam  lap,  or  steam  clearance.  The  design  is  completed 
at  A,  B,  C  and  D  as  before,  taking  S  on  the  valve  lever,  which  travels 
equally  either  side  of  the  vertical  center;  a  rod  connects  this  with  the 
crank-end  lever  which  is  longer,  and  receives  the  reach  rod.  The  bridge 
is  made  the  same  as  for  the  short-cut-off  gear  but  the  valve  faces  are  less. 


VALVES  AND  VALVE  GEARS 


433 


The  displacement  diagram  is  practically  the  same  as  for  a  single- valve 
engine,  but  due  to  a  negative  angular  advance,  is  tilted  in  the  opposite 
direction;  if  the  gear  fails  to  trip,  cut-off  will  not  occur  until  well  on  the 
return  stroke. 


f// 

Steam  Clearance  5 
* 

FIG.  279. 

Assuming  the  valve  to  close  in  0.1  of  a  revolution  as  before,  release  of 
the  gear  would  occur  at  6  to  give  a  %  cut-off;  the  actual  maximum  cut- 
off would  therefore  be  much  greater,  nearly  full  stroke.  The.  actual 
opening  curve  would  be  afghjh.  The  opening  is  not  quite  as  quick  as 


:O.I7S5troke 

~ 


FIG.  280. 


before,  but  while  there  is  a  large  over-travel,  the  maximum  valve  travel 
is  not  as  great  as  with  the  short-cut-off  gear,  the  travel  when  the  valve 
is  closed  and  under  pressure  being  much  less.  The  large  over-travel  is 
necessary  for  a  quick  port  opening  and  may  even  be  increased  if  desired. 


2S 


434 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Releasing  Gear  Operation. — There  are  various  designs  of  releasing  gears 
used  by  different  builders,  but  the  principle  is  practically  the  same  in 
all,  with  one  exception  (see  Nordberg  Bulletin).  A  simple  sketch  (not 
an  actual  design)  will  be  used  to  illustrate  the  principle  and  this  is  given 
in  Fig.  281,  which  is  similar  to  the  gear  used  on  the  Fishkill  Corliss 
engine.  This  type  is  used  as  the  operation  is  clearly  seen. 

The  valve  lever  or  steam  crank  a  is 
keyed  to  the  valve  stem.  To  it  is 
attached  the  dashpot  rod  6  which  pulls 
the  lever  down  and  closes  the  valve 
when  it  is  released.  The  bell  crank  (or 
steam  rocker)  c  receives  a  constant 
motion  from  rod  d  which  connects  with 
the  wrist  plate.  At  the  upper  end  of 
the  bell  crank  is  hung  the  latch  e,  hav- 
ing a  hardened  steel  block  at  its  lower 
end,  which  engages  with  a  like  block  on 
the  end  of  the  valve  lever.  The  latch 
is  secured  to  a  pin  which  passes  through 
a  hub  at  the  end  of  the  bell  crank,  and 
at  the  other  end  is  attached  a  lever, 
sometimes  called  the  knock-off  lever, 
which  carries  the  roller  /;  if  this  roller  is 
forced  away  from  the  valve  stem  the 

latch  is  unhooked.  The  cam  collar  g,  sometimes  called  the  governor  toe 
collar,  moves  independently  of  the  other  mechanism  and  is  attached  to 
the  governor  bell  crank  i>y  the  rod  h.  The  cam  collar  carries  two 
rollers,  i  being  the  knock-off  cam  and  j  the  safety  cam;  as  bell  crank  c 
oscillates,  the  latch  roller  /  may  come  in  contact  with  one  of  the  cam 
rollers,  depending  on  the  position  of  the  governor  when  the  latch  is 
unhooked.  For  simplicity  it  will  be  assumed  that  the  latch  is  unhooked 
when  the  centers  of  the  rollers  are  on  the  same  radial  line  from  the  center 
of  the  valve  stem. 

Fig.  282  is  a  diagram  showing  governor  and  cam  roller  in  the  three 
important  positions.  A  is  the  safety  position,  B  the  starting  position  and 
C  the  zero  cut-off  position.  These  will  be  taken  up  separately.  The 
latch  roller  has  a  constant  oscillation  through  angle  B. 

The  safety  position  A.  The  governor  is  in  its  lowest  position.  The 
latch  is  held  away  from  the  valve  lever  by  the  safety  cam  and  will  not 
"hook  up."  The  valves  remain  shut  and  the  engine  will  not  start,  or 
if  running,  will  stop.  To  start  the  engine  the  governor  must  be  lifted 


FIG.  281. 


VALVES  AND  VALVE  GEARS 


435 


so  that  it  rests  on  the  high  part  of  the  safety  collar,  or  more  safely,  on 
a  safety  pawl  which  drops  out  of  the  way  when  the  engine  gets  up  speed 
and  the  governor  lifts. 

The  starting  position  B.  The  valves  will  hook  up  and  will  not  un- 
hook, as  both  cut-off  and  safety  cams  are  out  of  reach  of  the  latch  roller. 
Cut-off  will  be  late,  depending  on  the  angular  advance.  When  the  engine 
speeds  up  and  the  governor  begins  to  take  control,  it  lifts  from  the  safety 
pawl  (shown  by  separate  sketch),  which  drops  out  of  the  way,  allowing 


ut-Off  Cam  on  Cam  Collar 
•Rollar  on  Latch 


^—Safety  Cam  on  Cam  Collar 
(Motneccessary  on  low  pressure 
cylinder  of  compound  eng/ne) 


Head-End  Valve  Diagram 
6-  Full  Angular  Movement 
of  Valve  Lever 


FIG.  282. 

the  governor  to  sink  to  its  safety  position  should  the  belt  break.  The 
so-called  safety  collar  which  must  be  turned  to  the  safety  position  by  hand 
after  the  engine  is  started  is  more  convenient  when  stopping  the  engine, 
but  is  only  a  partial  safety  device. 

Zero  cut-off  position  C.  The  latch  is  held  away  from  the  valve  lever 
by  the  knock-off  cam  and  will  not  hook  up.  This  is  a  limiting  position 
and  is  never  reached  unless  the  governor  "  hunts, "  as  the  cut-off  must  at 
least  be  long  enough  to  carry  the  friction  load. 

The  governor  in  action  gives  any  cut-off  between  zero  and  the  maxi- 
mum cut-off  under  governor  control.  Actual  cam  positions  may  be  found 


436  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

for  a  given  cut-off  which  is  to  be  the  same  at  head  and  crank  ends.  For 
equal  cut-offs,  the  knock-off  and  safety  cams  would  be  closer  together 
at  the  head  than  at  the  crank  end  if  it  were  necessary  to  have  the  safety 
cam  in  contact  at  the  extreme  lowest  position  of  the  latch  roller;  for, 
as  the  piston  moves  farther  on  its  stroke  from  head  to  crank  end  than  on 
the  return  stroke  for  the  same  movement  of  the  eccentric,  cut-off  would 
occur  later  in  the  head-end  than  in  the  crank-end  stroke  if  latch  roller 
and  knock-off  cam  bore  the  same  relation  at  both  ends  of  the  cylinder. 
Then  to  equalize  a  given  grade  of  cut-off,  the  head-end  cam  must 
be  nearer  to  the  latch  roller,  or  the  crank-end  cam  farther  away,  or 
both. 

The  safety  cam  should  always  keep  the  latch  from  hooking  up  when 
the  governor  is  in  the  lowest  position,  but  the  engagement  with  the  latch 
roller  need  not  be  at  the  lowest  position;  the  valve  lever  may  move  a 
certain  distance  before  the  valve  begins  to  open,  and  the  release  of  the 
latch  may  occur  at  any  point  within  this  distance,  which  is  much  less 
for  the  long-range  cut-off  than  for  the  single- wrist-plate  gear  as  may 
be  seen  by  comparing  Figs.  277  and  279. 

Comparing  the  long-  and  short-range  gears,  the  same  governor 
motion  (and  consequently  the  same  speed  fluctuation)  is  used  for  both; 
then  for  the  same  range  of  load  it  is  obvious  that  regulation  will  be  closer 
for  the  long-range  gear;  this  is  true  of  any  type  of  valve  gear. 

In  cross-compound  engines  the  low-pressure  cut-off  may  either  be 
adjusted  by  hand  to  any  fixed  position,  or  be  under  the  control  of  the 
governor.  In  the  latter  case  the  receiver  pressure  is  kept  more  nearly 
constant  under  change  of  load,  avoiding  very  low  pressures  with  light 
loads  and  excessively  high  pressures  with  heavy  loads.  When  the  low- 
pressure  cut-off  is  controlled  by  the  governor,  the  fulcrum  pin  of  the  con- 
trolling lever  is  lengthened  into  a  cross  shaft  having  one  bearing  on  the 
governor  stand,  which  is  usually  located  on  the  high-pressure  engine 
frame,  and  the  other  on  a  stand  occupying  a  similar  position  on  the  low- 
pressure  frame  and  sometimes  called  the  " dummy."  A  lever  is  placed 
on  the  low-pressure  end  of  the  shaft,  with  rods  connecting  it  with  the 
cam  collars  of  the  low-pressure  gear.  This  lever  is  sometimes  constructed 
so  that  it  may  be  adjusted  while  the  engine  is  running,  dividing  the  load 
between  the  two  cylinders  as  desired.  It  is  sometimes  arranged  so  that 
the  governor  may  be  temporarily  disconnected  and  hand  adjustment  used 
for  starting  as  explained  in  Chap.  XIII. 

Figs.  247  and  248,  Chap.  XIX  show  safety  devices  used  in  connection 
with  Corliss  and  other  trip  gears,  and  are  explained  in  the  text. 

Corliss  Valve  Stems. — With  certain  assumptions  the  following  formula 


VALVES  AND  VALVE  GEARS 


437 


has  been  derived,  where  D  is  the  cylinder  diameter,  d  the  valve  diameter 
and  di  the  diameter  of  the  stem  at  the  smallest  part. 

di  =  0.03  \ffi~D  (25) 


The  maximum  steam  pres- 
sure p  may  be  taken  as 
100  Ib.  per  sq.  in.  for  low- 
pressure  cylinders  to  give 
practical  results. 

146.  High-speed   Cor- 
liss Gears. — This  term  is 
commonly  applied  to  non- 
releasing  gears  when  rock- 
ing   valves    and    a    shaft 
governor     are     employed. 
The  diagrams  are  much  the 
same   as  for  the  releasing 
gear,  these  giving  the  maxi- 
mum cut-off,  which  may  be 
about  %  stroke;  then  for 
the    shorter    cut-offs    the 
eccentric  swings  or  shifts, 
giving  smaller  valve  travel 
and    greater    angular    ad- 
vance.    The  steam  distri- 
bution  is    practically    the 
same  as  for  a  single  valve 
except  that  the  opening  is 
a  little  quicker.     Assuming 
the  same  diagram  as  Figs. 
277  and  278,  the  curve  m 
is  the  valve  opening  curve 
for  J£  cut-off;  it  indicates 
that  greater  over-travel  for 
the  maximum  cut-off  would 
be  desirable. 

147.  Mclntosh   and 
Seymour  Gear. — Formerly 
the   gridiron-valve    engine 
built     by    Mclntosh    and 
Seymour     had     expansion 


frfcEml     g 


438  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

valves  on  the  steam  valves,  but  in  the  Type  F  engines  these  are  omitted, 
greatly  simplifying  the  gear  and  still  giving  good  results.  End  and  side 
elevations  of  this  gear  are  shown  in  Fig.  283.  The  motion  for  the  steam 
valve  is  equivalent  to  one  wrist  plate  driving  another.  The  valve  move- 
ment is  very  small.  The  mechanism  is  operated  from  rocking  lay  shafts 


FIG.  284. — Lentz  valve  gear. 

which  receive  their  motion  from  eccentrics.  The  steam  eccentric  is 
attached  to  the  governor  and  is  offset  in  its  connection  with  the  lay 
shaft  for  the  purpose  of  obtaining  equal  cut-offs. 

148.  Cam  and  Eccentric  Gears. — In  these  gears  the  cam  is  used  in 
place  of  the  wrist  plate  and  for  the  same  purpose.  They  are  used  only 
with  poppet  valves,  their  advantage  being  that  they  are  free  to  move 


VALVES  AND  VALVE  GEARS  439 

after  the  valve  is  seated  without  interfering  with  it.  The  crank-eccentric 
diagram  of  Fig.  265  may  be  used  and  the  cam  designed  to  give  as  quick  an 
opening  as  practicable.  As  with  the  Corliss  gear,  if  the  lead  is  ignored 
in  drawing  the  crank-eccentric  diagram,  the  cut-off  will  be  some  longer 
than  the  diagram  indicates;  a  slightly  shorter  cut-off  may  be  taken  to 
offset  this  if  desired. 

Lentz  Gear. — The  eccentrics  of  this  gear  are  on  a  lay  shaft  driven  by 
miter  gears.  The  steam  eccentrics  (one  for  each  end)  are  connected 
with  the  governor,  being  shifting  eccentrics  with  constant  lead.  An  end 
section  showing  the  general  arrangement  is  given  in  Fig.  284.  In  laying 
out  the  crank-eccentric  diagram  for  the  steam  valves,  the  line  of  stroke 
would  be  assumed  as  a  line  connecting  the  lay  shaft  center  with  the  mean 
position  of  the  cam  arm;  the  dead -center  crank  position  for  correspond- 
ing ends  of  the  cylinder  would  lie  along  this  line  toward  the  cam.  For  the 
exhaust  valves,  the  dead-center  position  would  be  on  the  side  opposite 
the  cam.  If  the  engine  is  right  hand  and  runs  over,  the  rotation  for  Fig. 
284  is  counterclockwise. 

149.  Reversing  Gears. — Space  will  not  permit  an  extended  treatment 
of  this  subject,  but  a  brief  description  of  a  few  of  the  most  important 
gears  will  be  given,  with  some  explanation  of  the  principles  involved; 
for  a  description  of  other  types,  or  a  fuller  treatment  of  any  given  type 
the  reader  is  referred  to  the  works  named  at  the  end  of  the  chapter.  Most 
of  the  illustrations  are  diagrammatic. 

Reversing  gears  are  usually  associated  with  the  locomotive  and  ship 
engine;  they  are  also  used  for  hoisting,  rolling  mills  and  for  some  other 
stationary  engines.  They  are  sometimes  used  in  connection  with  the 
Corliss  gear,  in  which  case  their  function  is  reversing  only;  but  in  most 
cases  they  are  also  used  for  varying  the  cut-off.  This  latter  function 
was  probably  discovered  accidentally.  As  first  applied  to  locomotives, 
reversing  gears  consisted  of  two  eccentrics,  each  with  its  own  eccentric 
rod  with  sort  of  a  crab-claw  hook  on  the  other  end;  either  one  of  these* 
could  be  hooked  onto  the  rocker  pin  by  a  double  bell  crank  operated  from 
the  cab;  this  gave  full  gear  forward  or  backward.  It  is  obvious  that 
the  reversing  operation  could  not  well  be  done  when  the  locomotive  was 
under  much  motion,  and  it  was  probably  to  obviate  this  defect  that  the 
sliding  link  of  the  Stephenson  gear  and  the  sliding  block  of  the  Walschaert 
gear  were  devised,  both  of  which  were  invented  about  the  year  1844. 

Most  reversing  gears  give  an  effect  similar  to  a  single  shifting  or 
swinging  eccentric,  and  when  the  path  of  change  can  be  approximately 
determined,  the  Bilgram  diagram  may  be  used  for  preliminary  work  if  a 
single  valve  is  employed. 


440 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Stephenson  Reversing  Gear. — This  is  perhaps  the  best  known  and  the 
most  difficult  to  lay  out,  as  there  are  many  variables  which  effect  the 
results.  Fig.  285  shows  a  link  which  has  been  used  on  stationary  engines; 
the  proportional  figures  are  based  upon  the  diameter  d  of  the  saddle  pin, 
which  may  be  made  0.1  the  cylinder  diameter  (Ds,  Chaps.  XII  and  XIII) 
for  unbalanced  valves  and  %  of  this  for  balanced  valves.  A  is  the  saddle 
with  pin  forged  integral,  by  which  the  link  is  lifted  and  lowered;  while 
B  is  the  link  block  which  is  drilled  and  counter  bored  for  the  pin  which 
is  fitted  to  rocker  or  valve-rod  crosshead.  For  solid-ended  link  as  shown, 
one  flange  of  the  link  block  is  loose,  and  fastened  to  the  block  with  rivets 
or  countersunk  screws. 


00 


25- \ 


1.25  <— 


FIG.  285. — Stephenson  link. 
»  . 

Fig.  286  shows  the  gear  in  neutral  position  (mid-gear)  at  A,  forward  gear 
at  B  and  backward  gear  (reverse)  at  C,  as  arranged  for  a  locomotive  with 
outside  admission  and  the  link  block  attached  directly  to  the  valve-rod 
crosshead.  Should  there  be  inside  admission,  or  a  rocker  which  reverses 
the  valve  motion,  the  crank  would  be  on  the  opposite  dead  center.  As 
it  is  locomotive  drafting  room  practice  to  show  the  cylinder  at  the  right, 
this  has  been  assumed  in  Fig.  286,  although  for  all  previous  work  it  has 
been  at  the  left. 

Equivalent  Eccentric. — The  link  radius  is  usually  equal  to  the  distance 
from  the  shaft  center  to  the  center  of  the  link  block  at  mid  travel.  In 


VALVES  AND  VALVE  GEARS 


441 


Fig.  287  assume  the  link  to  be  straight  and  the  eccentric  rods  so  long  that 
their  angularity  has  no  effect;  further  assume  that  the  valve-rod  cross- 
head  is  raised  and  lowered  instead  of  the  link.  Let  the  valve  rod  be 


Valve  Rod 


FIG.  286. 


moved  to  position  B,  %  of  the  distance  to  the  lower  end  of  the  link. 
Now  let  the  crank  and  eccentric  be  rotated  clockwise  through  a  small 
angle,  moving  the  end  of  the  link  the  distance  a;  the  valve  rod  would  then 


Valve  flod 


IB  Valve  Pod 


FIG.  287. 


be  moved  the  distance  a/2.  If  the  two  eccentrics  are  connected  by  a 
line,  it  is  clear  that  a  point  located  Y±  of  the  distance  from  top  to  bottom 
of  this  line  would  travel  a  horizontal  distance  of  a/2.  The  movement  of 


442 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


the  valve  rod  due  to  two  eccentrics  and  a  link  is  then  equivalent  to  that 
produced  by  a  single  eccentric  which  shifts  along  this  connecting  line 
in  the  same  ratio  that  the  link  block  is  shifted. 

Open  and  Crossed  Rods. — If  the  rods  are  open  when  the  eccentrics  are 
both  on  the  side  toward  the  link,  they  are  known  as  open  rods.  If  crossed 
when  in  this  position  they  are  known  as  crossed  rods;  this  is  shown  in 


Open  Rods 


Crossed  Rods 


FIG.  288. 


Fig.  288.  Placing  the  eccentrics  in  the  opposite  position  indicated  by 
the  dotted  lines  gives  the  center  of  the  link  its  maximum  travel  6;  this 
is  the  valve  travel  for  mid-gear,  or  when  the  equivalent  single  eccentric 
has  an  angular  advance  of  90  degrees.  From  Fig.  288  it  is  obvious  that 
due  to  the  effect  of  angularity,  b  is  greater  than  a  for  the  open  rods  and  less 
than  a  for  the  crossed  rods;  it  may  also  be  seen  that  when  the  crank  is 
on  the  dead  center  in  each  case  the  valve  displacement  for  full  gear  is  a/2 
and  for  mid-gear  it  is  6/2.  Assuming  the  gear  to  be  so  designed  that  full- 


Open  Rods  'Crossed  Rods 

FIG.  289. 

and  mid-gear  lead  are  the  same  at  both  ends  of  the  stroke,  this  may  be 
shown  on  the  Bilgram  diagram  as  in  Fig.  289.  The  curve  of  eccentric 
change  is  unknown  but  has  been  assumed  as  the  arc  of  a  circle. 

It  will  be  noticed  that  for  open  rods  the  lead  increases  from  full  to 
mid-gear;  this  is  always  used  for  locomotives  and  is  ideal  for  the  service 
as  it  gives  a  greater  port  opening  when  the  gear  is  "notched  up"  to  give 
short  cut-off  at  high  speed.  Equalization  of  lead  is  also  more  important 
at  mid-gear  than  at  full  gear  for  the  same  reason. 


VALVES  AND  VALVE  GEARS 


443 


Crossed  rods  give  a  decreasing  lead  which  may  be  zero  or  even  negative 
at  mid-gear;  this  is  sometimes  used  for  hoisting  engines  and  tractors 
when  it  is  desired  to  control  the  engines  by  the  reverse  lever;  if  placed 
in  mid-gear  no  steam  is  admitted. 


Lead 


FIG.  290.— Walschaert  gear.    - 

The  most  complete  practical  treatment  of  the  design  of  the  Stephenson 
link  motion  with  which  the  author  is  acquainted  is  given  in  Link  and 
Valve  Motion  by  W.  S.  Auchincloss. 

Walschaert  Reversing  Gear. — This  gear  has  largely  replaced  the  Steph- 
enson gear  in  locomotive  construction  in  the  United  States.  This  is 


JO 

Main  Rod 

Crossheadl 
Arm  ~~7  * 

i      \ 

Inside  Admfssion 


ionL'mk 


FIG.  291.  —  Walschaert  gear  for  inside  admission. 

largely  due  to  constructional  purposes,  although  it  possesses  some 
other  advantages.  Fig.  290  shows  a  general  view  of  the  gear  with  the 
crank  on  crank-end  dead  center  and  the  radius  rod  in  mid-gear  position. 
This  is  for  outside  admission  which  is  used  for  flat  valves.  Inside  ad- 


444 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


mission  is  used  with  piston  valves  and  requires  a  different  arrangement. 
The  radius  rod  is  dropped  to  the  lowest  position  for  full  forward  gear; 
this  divides  the  force  required  to  move  the  valve  between  the  link  trunnions 
and  the  eccentric  rod.  For  back  gear  the  radius  rod  is  above  the  trunnions, 
which  carry  the  combined  loads  of  radius  rod  and  eccentric  rod,  which, 
with  the  exception  of  switching  engines  is  for  a  small  proportion  of  the 
time.  Figs.  291  and  292  are  diagrams  showing  the  arrangement  for 
inside  and  outside  admission  respectively,  with  the  crank  on  the  head- 
end dead  center. 

Equivalent  Eccentric. — It  may  be  seen  that  the  eccentric  has  no  angular 
advance;  the  effect  of  angular  advance  is  obtained  by  the  combination 


FIG.  292.— Walschaert  gear  for  outside  admission. 

lever,  which,  when  the  radius  rod  is  in  mid  position  moves  the  valve  twice 
the  sum  of  lap  and  lead.  The  method  of  proportioning  this  lever  is 
obvious.  It  is  also  plain  that  the  effect  of  the  crosshead  might  be  ob- 
tained by  an  eccentric  with  an  angular  ad vance  of  90  degrees  for  an  outside- 
admission  gear  as  shown  at  B,  Fig.  293.  The  eccentric  which  operates 
the  link  is  at  A.  Connect  these  two  points  by  a  straight  line;  on  this 
line  a  point  C  will  be  located  which  depends  upon  the  proportions  of  the 
combination  lever,  which  in  turn  is  determined  by  the  lap.  Then  C  is 
the  equivalent  eccentric  which  would  give  the  same  results  as  the  gear. 
When  the  reverse  lever  is  " notched  up"  the  radius  of  eccentric  A  (which 
has  no  angular  advance)  is  virtually  decreased — say  to  OD.  As  the  lead 
is  constant,  C  must  travel  parallel  to  line  OD  to  the  new  position ;  at  any 


VALVES  AND  VALVE  GEARS 


445 


rate  the  ratio  AC/AB  is  constant,  so  C  must  move  in  a  vertical  line. 
As  C  moves  to  the  center  line,  eccentric  A  has  no  effect  on  it;  the  angular 
advance  of  the  equivalent  eccentric  is  90  degrees  and  the  valve  travel  is 
twice  the  sum  of  lap  and  lead.  As  C  crosses  the  line  toward  A\,  the  gear 
is  reversed.  This  may  be  shown  on  the  Bilgram  diagram  as  in  Fig.  294. 


FIG.  293. 


FIG.  294. 


Variable  lead  is  sometimes  used  on  passenger  and  fast  freight  locomo- 
tives equipped  with  the  Walschaert  gear.  It  is  obtained  at  the  expense 
of  too  great  a  lead  in  back  gear,  but  this  is  not  much  used  on  these  engines. 
Maximum  mid-gear  lead  is  determined,  which  may  be  greater  if  the  lead 
is  to  be  variable.  The  eccentric  is  then  set  back  for  forward  gear  by 


•\ 


FIG.  295. 


swinging  the  eccentric  crank  out  or  in,  depending  upon  whether  it  leads 
or  follows  the  crank,  thus  reducing  the  full-gear-forward  lead  and  in- 
creasing the  full-gear-backward  lead.  This  may  be  shown  on  a  Bilgram 
diagram  as  in  Fig.  295.  This  also  gives  a  longer  cut-off  and  later  com- 
pression in  forward  gear  upon  starting,  but  shorter  in  back  gear. 


446 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Hackworth  and  Marshall  Gears. — A  simple  diagram  of  the  Hackworth 
gear  is  shown  in  Fig.  296.  A  block  slides  in  the  straight  link  which  may 
be  turned  through  angle  A,  which  reverses  the  engine.  If  placed  in  a 

vertical  position  the  link  has  no  effect 
and  the  valve  travels  twice  the  sum 
of  lap  and  lead.  The  link  and  slid- 
ing block  are  equivalent  to  an  eccen- 
tric fixed  at  right  angles  to  the  crank 
(with  no  angular  advance);  there  is 
therefore  a  similarity  of  principle 
to  the  Walschaert  gear.  In  the 
Marshall  gear  the  link  and  block  are 
replaced  by  a  swing  link  as  shown 
in  Fig.  297.  Figs.  296  and  297  are 
for  outside  admission;  for  inside 
admission  the  eccentric  must  be  on 
the  same  side  of  the  circle  as  the 
ciank,  or  the  eccentric  rod  must  be 


FIG.  296.— Hackworth  gear. 


produced  beyond  the  link  block  and  the  rod  attached  to  this  extension. 
The  Hackworth  gear  is  used  on  the  Case  steam  tractor.  The  Marshall 
gear  has  been  used  on  reversing  Corliss  rolling  mill  engines  of  large  power. 


Line  of  Stroke 


FIG.  297.— Marshall  gear. 

Joy  Reversing  Gear. — In  this  gear  no  eccentric  is  used,  but  the  motion 
is  taken  from  the  connecting  rod.  A  link  is  attached  to  the  rod  at  A 
and  another  to  the  frame  at  B]  these  connect  at  C.  To  the  link  AC 


VALVES  AND  VALVE  GEARS 


447 


another  link  is  pivoted  at  D;  this  is  equivalent  to  the  eccentric  of  the 
Marshall  gear,  the  remainder  of  the  gear  being  much  the  same.  The 
arrangement  is  equivalent  to  having  the  eccentric  on  the  same  side  of  the 


Valve  Poof 


FIG.  298. — Joy  gear. 


FIG.  299. — Doble  gear. 

shaft  as  the  crank,  therefore  for  outside  admission  the  reversing  link  must 
be  between  the  point  D  and  the  valve  rod. 

The  Hackworth,  Marshall  and  Joy  gears  have  constant  lead;  but 
variable  lead  could  be  obtained  as  with  the  Walschaert  gear  by  causing 
the  angle  between  the  center  line  of  action  of  the  eccentric  rod  and  the 


448 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


line  of  stroke  to  be  slightly  different  from  90  degrees.     The  Bilgram  dia- 
gram for  these  gears  is  the  same  as  for  the  Walschaert  gear. 

The  Doble  reversing  gear,  used  on  the  Doble  steam  car,  is  like  the  Joy 
gear  with  links  AC  and  CB  omitted,  the  eccentric  rod  being  attached  at 
A  on  the  connecting  rod.  The  link  block  slides  in  a  straight  line,  being 
like  the  Hackworth  gear  in  this  respect;  it  is  practically  the  same  as  the 
Hackworth  gear  with  the  eccentric  replaced  by  the  connecting  rod  attach- 
ment at  A,  Fig.  299.  The  valve  has  inside  admission.  The  Doble  engine 
is  a  uniflow  engine,  so  that  the  gear  controls  the  steam  valves  only. 

INTERNAL-COMBUSTION  ENGINES 

150.  Cam  Gears. — General  arrangements  of  cam  gears  may  be  seen  in 
Chap.  V.  The  design  of  cams  in  general  may  be  found  in  books  on  mech- 
anism, but  application  will  be  made  to  two  cases.  The  first  thing  to  be 


Cylinder    v     1H\    - 


FIG.  300. 


determined  is  the  valve  timing,  and  Fig.  300  gives  the  position  of  the 
crank,  assuming  a  horizontal  engine.  Fig.  301  shows  the  timing  laid 
off  on  rectified  crank  circles,  two  revolutions  being  required  for  the  4- 


cycle  engine.  The  letters  H  and  C  denote  head  and  crank  ends  of  the 
stroke  respectively.  Fig.  302  shows  a  timing  diagram  referred  to  the 
cam  shaft,  being  curves  of  cam  rise  on  a  rectified  zero  circle  of  the  cam. 


VALVES  AND  VALVE  GEARS 


449 


As  the  cam  makes  one  revolution  per  cycle,  the  angles  e  and  i  are  one- 
half  the  angles  E  and  /  respectively.     It  may  be  assumed  that  the  cam 


Expansion              £>/ 
>°                        90° 

'  Zo* 

FIG.  302. 

L/on    '         Compress/or 

C     n                                    H 

270°                      36 

roller  travels  in  the  direction  of  the  arrow,  the  letters  H  and  C  indicating 
the  corresponding  positions  of  the  crank. 

Table  66  gives  values  of  E  and  /  taken  from  various  sources.     The 


FIG.  303. 
TABLE  66 


Case 

Eo 

EC 

Io 

Ic 

Roberts'  Handbook  

45  3 

10.3 

14.7 

35  4 

Large,  slow-speed  

25  0 

0  0 

5  0 

10  0 

Small,  medium-  or  slow-speed  

25.0 

0.0 

3.0 

4.0 

Small,  high-speed  

35  0 

2  0 

6.0 

15  0 

Bruce-Macbeth 

45  0 

15  0 

10  0 

45  0 

Continental  4-cylinder  truck  
Continental  6-cylinder  auto  

42.6 
55  0 

8.3 
12.0 

17.9 
12.0 

29.4 
45.0 

Racing  engine 

50  0 

10  0 

10  0 

50  0 

Low-speed  (Power,  Oct.  13,  1914)  
High-speed  (Power,  Oct.  13,  1914)  

30.0 
40.0 

5.0 
10.0 

10.0 
15.0 

10.0 
to 
15.0 
30.0 
to 

35.0 

29 


450 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


negative  sign  indicates  that  the  event  occurs  after  the  dead  center  at 
which  it  is  theoretically  supposed  to  occur. 

The  method  of  drawing  the  piston  velocity  curve  is  given  in  Chap. 
XVI ;  the  radius  of  the  crank  circle  may  be  taken  equal  to  the  rise  of  the 
cam  roller,  giving  a  curve  of  proper  scale  to  be  transferred  to  the  cam 
layout  for  comparison.  This  is  shown  in  Fig.  303. 

Using  Fig.  302,  as  a  guide,  Fig.  304  is  drawn  for  the  exhaust  cam  with 
roller.  The  cam  rise  is  denoted  by  h,  the  radius  of  the  base  circle  of  the 
cam  by  R  and  the  radius  of  the  roller  by  r.  The  following  proportions 
are  used  in  Fig.  304: 

R  =  2h,  r  =  0.6#  =  l.2h. 

It  may  be  assumed  that  the  cam  roller  rotates  clockwise  while  the 
cam  is  stationary.  The  engine  crank  passes  through  positions  H  and  C 


FIG.  304 — Exhaust  cam  with  roller. 


FIG.  305. — Inlet  cam  with  roller. 


(Fig.  303)  coincident  with  the  passing  of  the  center  of  the  cam  roller 
through  positions  so  marked  on  Fig.  304.  If  a  separate  spring  is  used 
to  return  the  push  rod  and  other  mechanism  transmitting  motion  to  the 
valve  stem,  the  cam  roller  rides  on  the  cam  body  of  radius  R,  the  clearance 
k  being  between  the  push  rod  and  valve-stem  lever.  According  to  G.  W. 
Meunch  in  Power,  Oct.  13,  1914,  k  should  be  about  ^2  in->  and  somewhat 
more  for  large  engines;  the  clearance  should  be  adjusted  when  the  engine 
is  hot.  For  small  engines  with  very  high  speed  the  clearance  is  some- 
times made  less. 

In  drawing  the  cams  the  center  of  the  roller  is  placed  at  the  opening 
and  closing  angles  as  shown;  the  cam-rise  lines  are  then  drawn  tangent 
to  the  base  circle  and  roller  circle  as  shown.  The  circular  arc  forming  the 
crown  of  the  cam  is  then  drawn,  and  joined  to  the  tangent  lines  with  a 
radius.  This  curve  may  be  transferred  to  the  roller-center  circle  as 


VALVES  AND  VALVE  GEARS 


451 


shown.  The  velocity  curve  may  be  laid  off  on  this  circle,  care  being 
taken  to  place  the  crank-end  of  the  curve  (the  low  end)  at  the  point 
marked  C.  If  desired  the  velocity  curve  may  be  transferred  to  the 
base  circle  as  shown  by  heavy  dotted  line.  Curves  A  are  the  velocity 
curves. 

It  may  be  seen  that  the  clearance  k  influences  the  form  of  the  cam, 
a  small  clearance  causing  the  valve  to  open  and-  close  more  slowly. 

Fig.  305  is  the  inlet  cam  drawn  in  the  same  manner.  Due  to  the 
opening  of  the  valve  after  dead  center  is  passed,  the  cam  curve  lies  below 
the  velocity  curve  at  first. 
The  cam  curve  B  tends  to  cor- 
rect this,  but  the  curve  shown 
at  D  is  the  curve  used,  as  the 
cost  of  production  is  less. 

A  type  of  cam  much  used 
for  small  engines  has  a  push 
rod  with  mush-room  head  with 
a  hardened  steel  face.  Ex- 
haust and  inlet  cams  are 
drawn  together  in  Fig.  306. 
The  form  of  the  cam  is  quite 
different,  and  the  opening  and 
closing  are  more  rapid.  The 
push  rod  is  held  against  the  cam  by  a  spring,  the  clearance  being  between 
the  push  rod  and  valve  stem  or  lever. 

Multi-cylinder  Gears. — The  usual  order  of  firing  for  multi-cylinder  en- 
gines is  given  in  Par.  106,  Chap.  XVI.  There  is  some  difference  of  opinion 
regarding  the  8-cylinder  engine.  Considering  cylinder  No.  1.  at  the  front 
of  the  car  and  the  driver's  right  as  R,  the  usual  order  of  firing  is:  1-L, 
2-#,  3-L,  l-R,  4-L,  3-R,  2-L  and  4-R.  The  Gas  Engine,  April,  1915,  gives 
for  the  "Ferro:"  1-L,  3-R,  3-L,  4-R,  4-L,  2-R,  2-L  and  l-R;  it  quotes 
the  reason  as  follows:  "(1)  Because  two  successive  impulses  acting  on 
one  crank  pin  has  substantially  the  effect  of  a  prolonged  impulse  with  the 
average  thrust  in  a  vertical  line,  and  since  all  shaft  and  crank-pin  bearings 
are,  or  should  be  split  in  a  horizontal  plane,  it  is  right  that  the  thrust 
should  be  as  nearly  as  possible  at  right  angles  to  the  path  of  the  split. 
(2)  If  the  spark  is  advanced  too  far,  and  No.  1  left  fires  too  early  after 
No.  1  right,  it  is  apparent  that  this  will  set  up  no  crank-shaft  strains 
whatever,  but  if  No.  4  left  followed  No.  1  right,  a  too  early  ignition 
means  an  opposed  torque  tendency  at  the  opposite  ends  of  the  crank 
shaft. " 


FIG.   306. — Inlet    and   exhaust   cams — mushroom 
type. 


452 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


151.  Eccentric-and-cam  Gear. — Trial  and  error  must  be  used  in 
determining  the  best  form  of  cam  and  the  proper  lift.  The  diagram  of 
Fig.  265  may  be  employed  for  the  crank  and  eccentric  as  seen  in  Fig.  307. 
The  eccentric  circle  may  be  assumed,  and  the  angles  determined;  then 
the  proper  dimensions  may  be  found  by  trial.  The  movement  of  the 
eccentric  rod  required  to  give  full  valve  lift  is  equal  to  h  when  the  proper 
scale  is  found.  H-0  is  a  reference  line,  or  center  line  of  an  imaginary 
crank,  revolving  at  the  speed  of  the  lay  shaft  (H  the  speed  of  the  engine 
shaft),  and  passing  through  points  marked  H  and  C  as  the  engine  crank 
passes  head-  and  crank-end  dead  centers  respectively,  thus  locating  the 
eccentric  position  for  any  position  of  the  engine  crank.  The  diameter 
H-H  lies  on  a  line  joining  the  lay-shaft  center  with  the  center  of  motion 


Exhaust 


Inlef- 


FIG.  307. 


of  the  cam  end  of  the  eccentric  rod,  the  side  on  which  h  lies  being  away 
from  the  cam  if  there  is  tension  in  the  eccentric  rod  in  opening  the 
valve. 

152.  Valve  springs  may  be  calculated  by  the  chart  and  formulas 
of  Par.  132,  Chap.  XIX,  using  PI  for  the  spring  tension  when  the  valve  is 
seated,  P%  when  wide  open  and  m  for  the  valve  lift  h,  or  the  spring  de- 
flection under  any  circumstances.     Further  let: 
z  =  weight  of  valve  and  other  parts  to  be  moved  by  spring. 
N  =  r.p.m.  of  engine. 

rT  =  ratio  of  time  required  to  close  valve,  to  time  required  per  stroke. 
Then  the  force  required  to  accelerate  the  valve  is: 

irhN  ,     . 

Pl  ~ 


VALVES  AND  VALVE  GEARS 


453 


The  force  required  to  hold  the  valve  against  suction  is: 


(27) 

where  D  is  the  cylinder  diameter  in  inches  and  p  the  maximum  suction 
in  Ib.  per  sq.  in.,  and  which  may  be  from  5  to  9  Ib.  The  larger  of  the 
two  values  of  PI  should  be  used.  A  good  value  of  Pz/Pi  is  about  1.15, 
but  this  may  be  increased  when  it  gives  a  larger  number  of  coils  than  is 
convenient. 

153.  Two-cycle  Engines. — The    piston    forms  the  valve  with   this 
type,  and  as  the  ports  are  opened  only  near  the  end  of  the  stroke,  the 
same  principles  do  not  apply  to  timing  and  port  opening.     It  has  been 
stated  that  the  width  (measured  along  the  stroke)  should  be  from  9  to 
13  per  cent,  of  the  stroke  and  the  exhaust  port  from  20  to  22  per  cent., 
and  each  should  extend  around  90  degrees  of  the  circumference.     This 
was  given  especially  for  heavy  oil  engines,  but  probably  applies  as  well 
to  gasoline  engines. 

154.  Reversing   Gears. — There  are  various  reversing  gears  used  on 
internal-combustion  engines,  especially  on  Diesel  engines  used  for  ship 


FIG.  308. — Diesel  engine  reversing  gear. 

propulsion.  Fig.  308  shows  one  type  used  on  Diesel  engines,  in  which 
there  are  two  cam  shafts,  either  of  which  may  be  in  gear. 

155.  Ignition  and  Fuel  Regulation. — High-tension  is  largely  used. 
In  this  type  the  sparking  and  timing  apparatus  is  not  usually  made  by 
the  engine  builder  so  space  will  not  be  taken  to  describe  it. 

With  low-tension  or  make-and-break  ignition  there  are  moving  parts 
connected  with  the  engine,  so  a  brief  description  will  be  given.  This 
type  is  used  for  large  gas  engines  and  sometimes  for  small  kerosene  engines. 


454 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


An  igniter  block  from  The  Gas  Engine  is  shown  in  Fig.  309  which  is 
almost  self  explanatory.  This  must  be  operated  by  some  form  of  trip 
mechanism  which  may  be  properly  timed.  Diagrams  of  two  methods 
are  given  in  Fig.  310.  Timing  may  be  changed  in  the  upper  method  by 

raising  or  lowering  the  roller  a  and  this  may 
be  under  governor  control  if  desired.  In  the 
lower  figure  the  cam  may  be  advanced  or  re- 
tarded. The  crank  6  and  cam  c  run  at  half 
the  engine  speed. 

Fuel  regulation  of  gas  engines  and  light-oil 
engines  is  commonly  done  by  throttling;  for 
the  latter  the  carbureter  is  used,  and  this  is 
not  usually  designed  or  manufactured  by  the 
engine  builder.  The  subjects  of  ignition  and 


FIG.  309. — Igniter  block. 


carburetion  are  too  extensive  to  attempt  a  treatment  ill  a  book  of  this 
kind  and  are  omitted.  For  gas  engines  the  regulation  usually  depends 
upon  a  mixing  valve  controlled  by  the  governor  and  this  is  shown  in 
Chap.  XIX. 


FIG.  310. 


The  Jacobson  Gas  Engine  Co.  use  a  form  of  trip  gear  and  this  is  shown 
in  Fig.  311. 

Oil  fuel  regulation  is  accomplished  in  several  ways.  Figs.  312  to  314, 
taken  from  Journal  of  A.S.M.E.  show  three  methods  used  with  Diesel 


VALVES  AND  VALVE  GEARS 


455 


FIG.  311. — Jacobson  releasing  gear. 


FIG.  312. — By-passing  to  suction  chamber.    FIG.  313. — Controlling  length  of  suction  stroke. 


456 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


engines.     In  all  cases  the  governor  rises  for  light  loads  when  less  fuel  is 
required;  the  principle  of  operation  may  be  easily  seen. 

A  regular  Diesel  spray  nozzle  is  shown  in  Fig.  315.  Oil  is  delivered 
under  pressure  between  perforated  plates  by  a 
small  passage;  when  the  fuel  valve  is  opened, 
high-pressure  air  enters  at  A  and  sprays  the 
oil  into  the  cylinder. 

An  open  nozzle  for  a  Diesel  engine  is  shown  in 
Fig.  316.  Oil  is  pumped  in  during  the  suction 
stroke  against  no  pressure;  it  is  then  sprayed 
into  the  cylinder  when  the  fuel  valve  is  opened. 
It  is  claimed  by  some  that  this  type  of  nozzle 
will  not  foul  as  easily  as  the  closed  nozzle. 

156.  The  Sleeve  Motor  Gear.— The  best 
diagram  for  designing  this  gear  is  doubtless  the 
rectangular  diagram  on  the  rectified  ciank  circle. 
Two  revolutions  of  the  crank  are  required.  The 
author  does  not  have  the  measurements  for  this 
engine,  so  the  diagram  of  Fig.  317  is  only  assumed. 
The  piston  displacement  curve  is  given  above, 
assuming  a  connecting  rod  4  cranks  long.  The 
valve-displacement  curves  neglect  the  angularity 
of  the  eccentric  rods,  but  this  may  easily  be 
accounted  for  if  desired.  Considerable  "cut- 
and-try"  is  necessary,  and  it  is  best  to  draw  the 
valve  curves  separately  on  tracing  cloth.  Valve- 
opening  curves  are  compared  with  piston  velocity  curves  (dotted),  laid 
off  on  a  rectified  crank  circle. 


FIG.  314. — Changing  stroke 
of  pump. 


FIG.  315. — Diesel  fuel  nozzle  and  valve. 


Different  arrangements  may  be  made  to  give  the  same  valve  timing; 
by  trial,  an  arrangement  may  be  obtained  which  will  give  the  best  results, 


VALVES  AND  VALVE  GEARS 


457 


and  this  has  doubtless  been  done  in  the  Willys-Overland  engines 
although  this  method  may  not  have  been  used.  Fig.  317  is  given  to 
show  its  utility  in  problems  of  this  kind. 


FIG.  316. — Open  type  of  fuel  nozzle. 


Exhaust  >  In  I 

FIG.  317. — Rectangular  diagram  for  sleeve  motor. 

157.  Details  from  Practice. — Some  idea  of  valve  gears  may  be  had  from 
Chaps.  Ill  and  V,  and  their  connection  with  the  governor  in  Chap.  XIX, 
but  a  few  details  will  be  given  in  this  paragraph. 


458  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIG.  318. — Eccentric  and  strap. 


FIG.  319. — Corliss  valves  and  stems. 


FIG.  320. — Bonnets  for  Corliss  valves. 


VALVES  AND  VALVE  GEARS 


459 


FIG.  321.— Bass-Corliss  dashpot. 


FIG.  322. — Lentz  poppet  valve. 


460 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Fig.  318  is  a  simple  fixed  eccentric;  the  eccentric  is  split  for  convenience 
in  placing  on  the  shaft,  and  the  strap  is  necessarily  split  for  placing  over 
the  eccentric,  and  it  also  serves  for  making  adjustment. 


</' 


FIG.  323. — Franklin  automobile  engine  valves. 

Fig.  319  shows  the  valves  and  stems  for  the  20  in.  Corliss  engine  for 
which  the  diagram  was  designed,  and  Fig.  320  shows  the  bonnets. 

Fig.  321  is  the  dashpot  used  by  the  Bass  Foundry  and  Machine  Co. 
It  is  adjustable  for  vacuum  and  cushion. 


FIG.  324. — Bruce-Macbeth  gas  engine  valves. 

Fig.  322  shows  a  poppet  valve  used  on  the  Lentz  super  heated-steam 
engine. 

Fig.  323  shows  the  valves  of  the  Franklin  Automobile  Engine,  and  Fig. 
324  and  Fig.  325  the  valves  and  cages  respectively  of  the  Bruce-Macbeth 
gas  engine. 


VALVES  AND  VALVE  GEARS 


461 


FIG.  325. — Bruce-Macbeth  valve  cages. 

References 

Slide  Valve  Gears F.  A.  Halsey. 

Valve  Gears W.  E.  Dalby. 

Valves  and  Valve  Gears F.  D.  Furman. 

Valve  and  Link  Motion W.  S.  Auchincloss. 

Handbook  and  bulletins  of  American  Locomotive  Company. 


PART  VI— MACHINE  DESIGN 

CHAPTER  XXI 
GENERAL  CONSIDERATIONS 

158.  Introduction. — In  this  chapter  are  discussions  of  a  few  of  the 
fundamentals  of  machine  design,  being  introductory  to  the  chapters  which 
follow,  obviating  the  necessity  of  interruption  in  the  way  of  derivation 
of  fundamental  formulas  before  they  may  be  applied  to  specific  cases. 

This  is  not  intended  as  a  treatment  of  the  mechanics  of  materials  or 
applied  mechanics;  on  the  other  hand,  some  knowledge  of  these  subjects 
is  assumed,  but  in  the  absence  of  such  knowledge  the  working  formulas 
given  may  be  used  without  the  ability  to  follow  their  derivation. 

In  the  use  of  all  formulas  in  machine  design  it  is  assumed  that  all 
material  is  homogeneous,  isotropic  and  free  from  internal  strains,  a  condi- 
tion of  things  never  fully  realized. 

In  this  chapter,  as  each  paragraph  deals  with  a  separate  subject,  the 
notation  does  not  apply  outside  the  paragraph  in  which  it  is  used,  al- 
though as  far  as  practicable  it  is  kept  uniform. 

159.  Rational  machine  design  consists  in  so  distributing  the  material 
used  in  machine  parts  that  economy  in  construction,  effectiveness,  safety 
and  durability  may  result.     This  does  not  of  necessity  imply  the  use  of 
rational  or  theoretical  formulas,  but  a  rational  application  of  the  formu- 
las which  most  correctly  express  the  behavior  of  materials  when  subjected 
to  the  loading  under  consideration.     Indeed,  we  may  go  a  step  farther 
in  the  case  of  machine  parts  in  which  the  acting  force  is  indeterminate,  or 
the  shape  of  the  section  such  that  a  correct  estimate  of  stress  relations 
is  impossible;  and  for  which  empirical  formulas  are  sometimes  used  which 
have  been  found  by  long  experience  to  give  proper  strength  and  rigidity. 
This  method,  when  used  by  experienced  and  skillful  designers,  can  hardly 
be  called  irrational.     However,  whenever  possible,  a  more  rigid  analytical 
treatment  is  more  satisfactory,  even  though  all  factors  involved  are  not 
accurately  known;  this  results  in  working  formulas  of  a  more  general 
character;  then  numerial  values  may  be  assumed  for  the  factors  which 
are  constant  under  a  given  set  of  conditions  (these  to  be  based  upon  ex- 

463 


464  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

perience  or  the  best  judgment  of  the  designer),  and  the  formulas  may 
often  be  reduced  to  a  single  constant  with  perhaps  one  or  two  variables. 
In  this  simple  form  the  constant  is  readily  compared  with  that  obtained 
from  successful  machines  of  the  same  class,  operating  under  similar  con- 
ditions; but  in  case  of  disagreement,  judgment  must  not  be  too  hasty  in 
favor  of  the  existing  machine. 

In  all  machine  design,  system  and  uniformity  of  practice  should  be 
maintained  From  time  to  time,  in  the  light  of  new  knowledge  gained 
by  experience  or  from  the  published  investigations  of  experts,  it  may  be 
necessary  to  change  values,  but  the  indiscriminate  and  capricious  vary- 
ing of  constants  by  designers  is  as  unsatisfactory  as  it  is  unscientific,  and 
leaves  no  reference  point  from  which  to  measure  progress. 

The  determination  of  proper  bearing  pressures  and  working  stresses 
is  of  paramount  importance  in  rational  design.  Bearing  pressures 
depend  entirely  upon  judgment  and  experience;  working  stresses  or  unit 
loads,  with  which  this  discussion  is  concerned,  have  been  determined 
partly  by  experience  and  partly  from  laws  deduced  from  tests  of  materials. 

By  working  stresses  and  unit  loads  are  meant  those  values  used  in 
formulas  for  design,  under  the  assumption  of  simple  loading  upon  which 
the  derivation  of  the  formula  rests.  They  are  properly,  in  many  cases, 
only  factors  of  design,  the  actual  stresses  differing  greatly  from  them. 
The  more  simple  the  loading  or  thorough  the  analysis,  the  more  nearly 
will  the  assumed  stress  approach  the  actual  maximum  stress. 

Working  Stresses  and  Factors  of  Safety. — It  has  never  been  considered 
safe  to  use  a  working  stress  just  inside  the  stress  causing  rupture,  so  a 
factor  of  safety  has  always  been  employed.  Formerly  this  was  an  arbi- 
trary value  and  often  took  no  account  of  the  manner  in  which  the  load 
was  applied.  It  is  now  customary  to  assume  working  stresses  which  have 
been  proven  by  practice  to  be  satisfactory.  These  are  usually  determined 
with  reference  to  some  property  of  the  material,  such  as  the  elastic  limit. 

The  statement  often  made  that  there  is  no  fixed  rule  governing  the 
selection  of  a  working  stress  is  as  unsatisfactory  as  it  is  true;  however, 
the  results  of  Wohler's  tests  on  the  fatigue  of  materials  under  various 
forms  of  loading  furnish  a  means  of  checking  the  values  of  working  stress 
in  common  use  in  a  partially  rational  manner  at  least.  These  experi- 
ments were  upon  wrought  iron  and  steel,  and  the  laws  deduced  therefrom 
may  be  considered  strictly  applicable  only  to  ductile  materials  having 
approximately  the  same  properties. 

Fatigue  tests  for  the  purpose  of  comparing  different  kinds  of  steel 
have  more  recently  been  made  in  which  the  stress  used  was  considerably 
beyond  the  elastic  limit,  actually  bending  the  test  pieces  to  a  point  of 


GENERAL  CONSIDERATIONS  465 

permanent  set  at  each  repetition  of  the  load.  These  tests  showed  the 
excellence  of  a  certain  steel  greatly  to  the  disparagement  of  other  high- 
grade  steels;  and  there  is  no  doubt  that  in  case  of  miscalculation  or  acci- 
dent, steel  showing  high  fatigue  values  under  these  tests  would  resist 
failure  under  such  abuse  better  than  that  with  which  it  was  compared ; 
but  while  there  is  a  feeling  of  safety  in  the  use  of  material  which  will 
best  withstand  the  most  vigorous  tests,  the  author  doubts  if  the  tests 
mentioned  alter  the  applicability  of  the  laws  deduced  from  the  older  tests, 
to  rational  design. 

From  a  discussion  of  many  experiments  on  repeated  stresses  by  Woh- 
ler  and  others,  Prof.  Merriman  (Mechanics  of  Materials)  states  the  fol- 
lowing laws : 

1.  The  rupture  of  a  bar  may  be  caused  by  repeated  applications  of  a  unit 
stress  less  than  the  ultimate  strength  of  the  material. 

2.  The  greater  the  range  of  stress,  the  less  is  the  unit  stress  required  to  produce 
rupture  after  an  enormous  number  of  applications. 

3.  When  the  unit  stress  in  a  bar  varies  from  zero  up  to  the  elastic  limit,  an 
enormous  number  of  applications  is  required  to  cause  rupture. 

4.  A  range  of  stress  from  tension  into  compression  and  back  again,  produces 
rupture  with  a  less  number  of  applications  than  the  same  range  of  stress  of  one 
kind  only. 

5.  When  the  range  of  stress  in  tension  is  equal  to  that  in  compression,  the 
unit  stress  that  produces  rupture  after  an  enormous  number  of  repetitions  is  a 
little  greater  than  one-half  the  elastic  limit. 

Prof.  J.  B.  Johnson  (Materials  of  Construction),  discussing  numerous  stress 
diagrams  for  steel,  reaches  certain  conclusions,  some  of  which  may  be  added  to  the 
laws  just  given,  and  are  as  follows: 

6.  The  " apparent  elastic  limit"  is  found  between  60  and  70  per  cent,  of  the 
ultimate  strength  in  tension. 

7.  The  "apparent  elastic  limit"  in  compression  is  practically  the  same  as  that 
in  tension. 

8.  The  ultimate  strength  in  compression  is  practically  equal  to  the  "apparent 
elastic  limit." 

Prof.  Johnson  refers  to  a  column  test  on  p.  360  of  Materials  of  Construction 
which  confirms  this  third  statement  (Law  8)  in  which: 

length  =  I  =  2Q 

least  radius  of  gyration      r 

All  authorities  do  not  agree  with  this  statement,  nor  do  all  tests  confirm 
it,  especially  for  smaller  values  of  l/r,  but  it  is  on  the  side  of  safety  and 
will  be  assumed  as  true  in  the  following  discussion. 

The  term  " apparent  elastic  limit"  used  by  Prof.  Johnson,  also  called 
by  him  the  "commercial  elastic  limit,"  refers  to  the  elastic  limit  in  ordi- 


466 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


nary  use,  in  distinction  from  the  "true  elastic  limit"  or  " limit  of  propor- 
tionality, "  which  is  somewhat  less  than  the  elastic  limit. 
Stress  is  produced  in  engine  parts  in  three  ways,  as  follows: 

1.  Static  stress,  produced  by  an  unchanging  load. 

2.  Repeated  stress,  produced  by  a  constant  repetition  of  all  or  a  part 
of  the  load,  producing  stress  of  one  kind,  tension  or  compression.     The 
limiting  case  is  expressed  by  Law  3,  the  stress  ranging  from  zero  to 
maximum. 

3.  Reversed  stress,  which  changes  from  tension  to  compression.     The 
limiting  case  is  given  by  Law  5,  in  which  the  tension  equals  the  compres- 
sion. 

Formulas  expressing  Wohler's  results  may  be  found  in  the  works 
already  referred  to.  The  limiting  cases  given  by  Laws  3  and  5  are  the 
most  important,  intermediate  conditions  being  few,  or  seldom  known  with 
accuracy  in  engine  design. 

Let  Su  =  the  ultimate  strength  of  the  material  in  Ib.  per  sq.  in. 
SE  =the  elastic  limit  in  Ib.  per  sq.  in. 

n  =  SE/SU- 

Then  from  Laws  3,  5  and  8,  the  stress  at  failure  for  the  three  conditions 
are  given  in  Table  67. 

TABLE  67 


Load 

Static 

Repeated 

Reversed 

Tension 

Compression 

Stress  

S0 

SM 

SE 

SB/2 

Good  practice  dictates  that  under  no  condition  should  the  members 
of  a  machine  or  structure  be  strained  beyond  the  elastic  limit;  so  making 
the  maximum  static  stress  in  tension  equal  to  the  elastic  limit,  and  re- 
ducing the  other  values  in  the  same  proportion,  we  have  the  values  of 
Table  68. 

TABLE  68 


Load 

Static 

Repeated 

Reversed 

Tension 

Compression 

Stress  

SB  =  nSu 

nSE 

nSE 

I 

GENERAL  CONSIDERATIONS 


467 


This  undoubtedly  places  the  live  load  stresses  well  within  the  limit  of 
proportionality.  Taking  the  elastic  limit  as  the  basis  of  stress  measure- 
ment for  ductile  materials,  in  which  this  property  is  as  well  defined  as 
any  other,  we  may  consider  the  first  factor  of  safety  /i,  as  the  ratio  of  the 
elastic  limit  to  the  values  given  in  Table  68.  For  tension,  with  different 
values  of  n,  f\  is  given  in  Table  69. 

TABLE  69 


Load 

Static 

Repeated 

Reversed 

n  — 

n 

1 

i/m 

2/n 

n  =• 

0.5 

1 

2.00 

4.00 

/i 

n  = 

0.6 

1 

1.67 

3.34 

n  = 

2/3 

1 

1.50 

3.00 

n  = 

0.7 

1 

1.43 

2.86 

Selecting  the  value  of  /i  when  n  =  %  gives  a  dead-load  factor  which 
will  bear  a  greater  ratio  to  the  live-load  factor  than  if  a  smaller  value  of  n 
were  assumed ;  and  since  /i  is  based  on  the  elastic  limit,  this  assumption 
is  more  apt  to  place  the  static  stress  within  the  elastic  limit  when  the  live- 
load  stresses  are  within  the  limit  of  proportionality.  Values  of  /i  for 
the  different  ways  of  producing  stress  are  given  in  Table  70. 

TABLE  70 


Load 

Static 

Repeated 

Reversed 

Tension 

Compression 

/i 

1 

1.5 

1.5 

3 

On  account  of  some  uncertainty  as  to  just  what  constitutes  the  elastic 
limit,  this  factor  /i  may  be  said  to  insure  stresses  within  the  point  of 
possible  failure  when  the  commercial  elastic  limit  is  used  as  a  basis.  If 
the  material  is  furnished  by  specification  or  the  properties  known  from 
test,  /i  should  provide  against  failure  for  the  applications  of  load  given. 
It  is  usually  better  to  increase  the  factor  somewhat,  as  the  properties  of 
any  grade  of  material  vary  more  or  less,  and  in  many  cases  the  material 
is  known  only  in  a  general  way  by  the  designer.  The  factor  /i  may  then 
be  multiplied  by  a  second  factor  f%.  For  general  engine  design  a  practical 
value  for  this  may  be  taken  as: 

/I    -2. 

In  cases  where  lightness  is  imperative,  or  where  strains  due  to  high  tem- 
perature might  be  increased  by  thicker  metal,  this  factor  may  be  de- 


468 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


creased,  approaching  unity  if  the  properties  of  the  material  are  well 
known. 

Uneven  distribution  of  stress,  initial  stress  due  to  tightening  nuts  or 
driving  keys  (and  greater  than  the  calculated  stress  caused  by  the  working 
load),  or  other  straining  actions  not  easily  calculated,  may  be  provided 
for  by  a  third  factor  /3. 

Suddenly  Applied  Loads. — If  the  full  load  were  applied  instantane- 
ously but  without  shock,  it  is  shown  in  treatises  on  applied  mechanics 
that  the  stress  produced  is  double  that  caused  by  the  same  load  gradually 
applied.  Prof.  Unwin  says  that  "  practical  cases  rarely  approximate  to 
these  conditions." 

Shock. — This  may  be  caused  by  lost  motion  being  suddenly  taken  up. 
Even  with  badly  worn  bearings  where  there  is  a  perceptible  " knock," 
it  is  probable  that  but  a  small  percentage  of  the  maximum  load  on  any 
part  is  applied  as  sudden  load  or  shock. 

Factor  of  Judgment. — Provision  for  sudden  load  and  shock,  and  for 
other  unknown  straining  actions  may  be  included  in  the  third  factor  /3 
just  mentioned.  It  is  obvious  that  this  is  more  nearly  a  factor  of  judg- 
ment and  experience  than  the  others. 

The  total  factor  of  safety  is  then  a  product  of  the  three  factors,  or: 

/ = /i/i/i  (i) 

The  factor  f\  may  be  assumed  as  a  fixed  minimum  limit;  fz  may  vary 
according  to  the  accuracy  with  which  the  elastic  limit  of  the  material  is 
known,  but  will  be  taken  as  2,  as  previously  stated,  for  ordinary  engine 
work.  The  product  of /i  and/2  may  be  taken  as  a  standard  factor  fA,  or: 

h  =  AA  (2) 

This  suits  all  cases  for  simple  applications  of  stress  given  in  Table  71. 

TABLE  71 


Load 

Static 

Repeated 

Reversed 

Tension                   Compression 

U 

2 

3 

3 

6 

For  the  cases  of  irregular  loading  just  given,  fA  must  be  multiplied  by 
the  factor  of  judgment /3,  or: 

/=/«/*  (3) 

In  the  design  of  engine  details  which  follows,  practical  working  stresses 
for  ordinary  conditions,  based  largely  upon  experience,  have  been  assumed ; 
from  these  the  total  factor  has  been  found,  and  by  using  fA  as  given  in 


GENERAL  CONSIDERATIONS  469 

Table  71,  the  value  of  /3  was  obtained.     If  S  is  the  working  stress  assumed 
and  SE  the  elastic  limit  : 


or: 


This  method  seems  to  place  the  cart  before  the  horse,  but  as  both  S 
and  /3  depend  upon  experience  and  judgment,  one  serves  as  a  check  upon 
the  other,  no  matter  which  is  first  selected. 

The  less  accurate  the  analysis,  the  greater  must  /3  be.  In  the  use  of 
formulas  which  err  largely  on  the  side  of  safety,  such  as  most  of  those  for 
flat  plates,  the  factor  of  judgment  may  be  reduced  ;  but  when  there  are 
several  theories  giving  results  differing  considerably,  as  in  the  case  of  com- 
bined bending  and  torsion,  the  selection  of  the  least  safe  formula  offset 
by  a  large  factor  of  safety  is  ill  advised. 

The  use  of  such  formulas  is  sometimes  based  upon  their  apparent 
agreement  with  tests  to  destruction,  but  working  loads  never  impose  these 
conditions  upon  the  material,  and  such  tests  are  no  measure  of  its  fatigue- 
resisting  properties  It  is  better  in  such  cases,  when  the  limit  of  elastic 
strength  cannot  be  determined  by  test,  to  adopt  the  safest  rational  for- 
mula, selecting  the  factor  of  safety  as  for  more  simple  cases  of  straining 
action. 

A  thorough  analysis  of  even  the  best  designs  would  disclose  deviations 
from  the  assumptions  made  in  the  derivation  of  the  formulas  and  selection 
of  working  stress.  Though  a  quantitative  analysis  is  often  impracticable, 
a  study  of  the  kind  of  strains  possible  through  a  lack  of  exact  conformity 
to  a  design  by  the  shop,  will  assist  in  making  assumptions  and  in  deter- 
mining the  nominal  working  stress;  but  no  matter  how  thoroughly  we 
study,  in  the  end  the  factor  of  safety  will  always  be  an  absolute  necessity, 
although  sometimes  considered  to  reflect  upon  the  ability  of  designers. 

In  applying  Table  71  to  beams,  the  factor  for  static  load  may  be  taken 
the  same  as  for  tension.  When  the  beam  is  long  and  unsupported  later- 
ally, there  are  a  number  of  formulas  which  have  been  devised  to  limit  the 
load.  This  comes  under  structural  design  but  is  worthy  of  consideration. 

Cast  iron  and  other  brittle  materials,  having  no  marked  elastic  limit, 
the  factor  of  safety  must  be  based  upon  the  ultimate  strength;  but 
shrinkage  strains,  and  the  possibility  of  unsound  castings  necessitate  t  he 
exercise  of  a  good  deal  of  judgment. 

Under  ordinary  conditions  a  factor  of  4  for  static  loads  may  be  con- 
sidered sufficient.  This  may  be  assumed  to  correspond  to  fA  just  found  for 


470  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

ductile  materials,  with  the  exception  that  the  ultimate  strength  will  be 
used  as  a  basis  for  cast  iron.  In  the  absence  of  experimental  data  for 
repeated  and  reversed  stresses,  the  same  ratio  of  factors  may  be  assumed 
as  for  the  ductile  materials.  Factors  for  cast  iron  are  given  in  Table  72, 
suitable  alike  for  tension  and  compression. 

TABLE  72 


Load 

Static 

Repeated 

Reversed 

u 

4 

6 

12 

The  factor  /3  may  also  be  applied  as  already  given. 

As  the  compressive  strength  of  cast  iron  is  at  least  five  times  as  great 
as  the  tensile  strength,  it  may  perhaps  be  permissible  to  assume  that  the 
compressive  stress  may  be  five  times  the  tensile  stress  when  the  reversal 
factor  fA  is  used,  based  on  the  tensile  stress.  Or  for  reversed  stress  of 
equal  intensity,  the  compressive  stress  may  be  considered  as  one-fifth  of 
its  actual  value.  Then  from  the  chart  of  Fig.  326,  this  would  give  a 
standard  factor  fA  of  7.2  instead  of  12. 

When  cast  iron  is  used  where  failure  might  be  attended  with  disaster, 
as  in  steam  cylinders,  the  factor  /3  should  be  increased  accordingly; 
on  the  other  hand,  high  temperature  in  gas  engine  cylinders  calls  for 
thinner  walls  with  a  consequent  higher  maximum  allowable  stress. 

The  materials  most  usually  subjected  to  stress  in  engines  are  forged 
machinery  steel,  steel  castings  and  iron  castings.  Wrought  iron  was 
formerly  much  used,  but  has  been  almost  entirely  superseded  by  steel, 
usually  in  the  form  of  forgings,  but  sometimes,  as  in  locomotive  frames, 
by  steel  castings.  The  elastic  limit  is  well  defined  in  steel  castings,  and 
factors  of  safety  may  be  applied  as  for  forgings. 

The  mechanical  properties  of  a  given  material  may  vary  with  the 
composition,  treatment,  form  and  size  of  specimen,  and  even  with  the 
method  of  testing.  They  are  not  always  the  same  in  tension  and  com- 
pression, but  except  in  the  case  of  cast  iron,  they  may  be  assumed  equal. 
The  values  given  in  Tables  73  and  74  are  safe  for  ordinary  engine  design. 

E  =  modulus  of  elasticity  in  tension  or  compression. 

Es  =  modulus  of  elasticity  in  shear. 

SE  =  elastic  limit  in  tension  or  compression. 
SES  =  elastic  limit  in  shear. 

Su  =  ultimate  strength  in  tension  or  compression. 
Sus  =  ultimate  strength  in  shear. 
SUR  =  modulus  of  rupture  in  bending. 

All  the  above  values  are  in  pounds  per  square  inch. 


GENERAL  CONSIDERATIONS 
TABLE  73 


471 


Material 

E 

as 

SE 

SES 

Wrought  iron  
Machinery  steel  

26,000,000 
29  000  000 

10,000,000 
11  200  000 

27,000 
38  000 

21,000 
29  000 

Steel  casting  .  . 

24.000.000 

9.  200.  000 

30.000 

92  OOO 

TABLE  74 


/ 

Su 

Es 

Tension 

Compres- 
sion 

Sus 

SUR 

Cast  iron  

12,000,000 

4,600,000 

16000 

90000 

18000 

24  000 

For  static  shearing  stress,  the  same  factor  of  safety  may  be  used  as  for 
tension. 

Higher  speeds  in  engines  of  older  type,  the  extremely  high-speed 
engines  used  in  automobiles,  airplanes,  etc.,  and  the  steam  turbine  have 
been  in  part  responsible  for  the  development  of  steels  of  higher  grade, 
and  the  properties  of  some  of  the  most  reliable  and  widely  used  will  be 
given  here. 

The  notation  is  as  follows: 


C  =  carbon 
Mn  =  manganese 
Si  =  silicon 
S   =  sulphur 


P  =  phosphorus 

Ni  =  nickel 

Cr  =  chromium 

V  =  vanadium 


Table  75,  from  the  Proceedings  of  the  American  Society  for  Testing 
Materials,  1905,  gives  specifications  for  carbon  and  nickel  steels.  C.A. 
denotes  carbon  steel  annealed;  C.O.,  carbon  steel,  oil  tempered;  N.A., 
nickel  steel,  annealed,  etc.  The  elastic  limit  in  Table  75  is  taken  as  that 
point  at  which  the  proportionality  changes. 

For  bending  tests,  a  specimen  1  by  %  in-  shall  bend  cold  180  degrees 
without  fracture  on  outside  of  bent  portion,  as  follows :  (a)  around  a  diam- 
eter of  J^  in.;  (6)  around  a  diameter  of  1  in.;  (c)  around  a  diameter  of 
1M  in.;  (d)  no  bending  test  required. 

Chemical  composition:  P  and  S  not  to  exceed  0.04  per  cent,  in  carbon 
or  nickel  steel,  oil  tempered,  or  0.05  per  cent,  in  locomotive  forgings. 
Mn  not  to  exceed  0.60  per  cent.  Ni  3  to  4  per  cent,  in  nickel  steel. 


472 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


TABLE  75. — SPECIFICATIONS  FOR  STEEL 


Steel  forgings 

Kind 
of 
steel 

SB 

Elongation 
in  2  in., 
per  cent. 

Reduc- 
tion in 
area, 
per  cent. 

Solid  or  hollow  forgings,  no  diameter  or  thickness 
to  exceed  10  in  

c 

C.A. 

37,500 
40,000 

18 
22 

30(cj 
35(6) 

Solid  or  hollow  forgings,  diameter  not  to  exceed 
20  in  ,  or  thickness  of  section  1  5  in 

N.A. 
C.A. 
N  A 

50,000 
37,500 
45  000 

25 
23 
25 

45  (a) 
35(6) 
45(01 

Solid  forgings,  over  20  in  

C.A. 

35000 

24 

30  (c) 

Solid  forgings  

N  A 

45  000 

24 

40  (a} 

Solid  or  hollow  forgings,  diameter  or  thickness 
not  over  3  in  

C.O. 

N  O. 

55,000 
65000 

20 
21 

45(6) 
50(6) 

Solid  rectangular  section,  thickness  not  over  6  in., 
or  hollow  with  walls  not  over  6  in.  thick  
Solid  rectangular  section,  thickness  not  over  10 
in.,  or  hollow  with  walls  not  over  10  in.  thick  .  .  . 
Locomotive  forgings  

C.O. 
N.O. 
C.O. 
N.O. 

50,000- 
60,000 
45,000 
55,000 
40,000 

22 

22 
23 
24 
20 

45(6) 
50(6) 
40(6) 
45(6) 
25  (d) 

Bulletin  100  of  the  Bureau  of  Mines,  entitled :  Manufacture  and  Uses 
of  Alloy  Steels,  by  Henry  D.  Hibbard,  gives  much  valuable  information. 
Tables  76  to  78  give  some  selected  values  from  this  source. 

W850,  A538  indicates  that  the  sample  was  quenched  in  oil  at  850° 
C.  and  the  hardness  drawn  in  air  at  538°  C.  O926  denotes  quenching 
in  oil  at  a  temperature  of  926°  C. 

Nickel  Steels. — As  stated  by  the  author  of  the  bulletin  mentioned, 
nickel  steel  containing  3.25  to  3.5  per  cent,  of  nickel  and  known  as 
ordinary  nickel  steel,  has  a  high  value  for  structural  purposes  such  as 
bridges,  gun  forgings,  machine  parts,  engine  and  automobile  parts,  and 
any  similar  line  of  service  too  severe  for  simple  steels.  It  has  been  stated 
that  alloy  steels  possess  no  advantage  over  simple  steels  if  not  heat 
treated,  but  that  the  alloys  may  even  have  a  deleterious  effect;  but 
nickel  steel,  when  used  in  bridge  work,  is  used  in  the  natural  or  annealed 
condition,  when  the  additional  strength  and  ductility  is  due  only  to  the 
presence  of  the  nickel. 

Table  76  gives  the  composition  and  properties  of  ordinary  nickel 
steel,  and  it  may  be  seen  that  it  may  be  made  suitable  for  any  structural 
purpose  for  which  it  is  not  too  expensive. 

Nickel-chromium  Steels. — These  steels  are  perhaps  the  most  important 
structural  alloy  steels,  and  their  field  of  application  is  continually  being 
enlarged.  They  are  seldom  used  in  any  but  a  heat-treated  condition. 
By  suitable  treatment  small  pieces  may  have  as  high  physical  properties 


GENERAL  CONSIDERATIONS  473 

TABLE  76. — PROPERTIES  OF  ORDINARY  NICKEL  STEEL 


Composition 

Elonga- 

Contrac- 

No. 

c, 

Mn, 

Si, 

s, 

P, 

Ni, 

Condition 

SE 

tion, 

tion, 

per 

per 

per 

per 

per 

per 

per 

per 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

1 

0.28 

0.57 

0.03 

0.02 

3.44 

Natural  state 

56,670 

21.2* 

50 

c\ 

0.40 

0.64 

0.02 

0.01 

3.43 

Annealed 

51,400 

12.  4f 

33 

3 

0.40 

0.55 

0.03 

0.01 

3.70 

Annealed 

56,060 

15.  8f 

40 

4  0.20 

0.65 

.... 

0.04 

0.04 

3.50 

Annealed 

43,000 

27.0 

62 

5  0.20 

0.65 

0.04 

0.04 

3.50 

W850,  A538 

95,000 

20.0 

72 

6 

0.20 

0.65 

0.04 

0.04 

3.50 

W800,  A316 

140,000 

14.0 

61 

7 

0.30 

0.65 

.... 

0.04 

0.04 

3.50 

Annealed 

63,000 

27.0 

63 

8 

0.30 

0.65 

0.04 

0.04 

3.50 

W800,  A593 

87,000 

25.0 

68 

9 

0.30 

0.65 

0.04 

0..04 

3.50 

W800,  A399 

123,000 

15.0 

57 

10 

0.30 

0.65 

.... 

0.04 

0.04 

3.50 

W800,  A316 

187,000 

13.0 

57 

11  0.25 

0.74 

0.21 

0.01 

0.01 

3.55 

W843,  A316 

177,000 

14.0 

60 

12  0.25 

1 

0.74 

0.21 

0.01 

0.01 

3.55 

W843,  A538 

117,000 

20.0 

67 

*In  8  in.     fin  18  ft. 

as  any  known  steel,  and  with  any  elastic  limit  from  40,000  to  250,000, 
accompanied  by  ductility  that  is  high  as  compared  with  strength,  the 
ductility  naturally  lessening  with  increase  of  elastic  limit. 

Nickel-chromium  steel  may  be  made  some  cheaper  than  simple 
nickel  steel  of  the  same  strength  and  ductility,  as  it  contains  a  smaller 
amount  of  the  alloying  elements,  which  are  also  less  expensive  than 
nickel.  Table  77  gives  the  properties  of  nickel-chromium  steels. 

TABLE  77. — PROPERTIES  OF  NICKEL-CHROMIUM  STEEL 


No. 

Composition 

Condition 

SE, 

Elongation 
in  2  in., 
per  cent. 

Contraction, 
per  cent. 

c, 

per 

cent. 

Mn, 
per 
cent- 

Si, 
per 
cent. 

s, 

per 

cent. 

P, 

per 
cent. 

Ni, 
per 
cent. 

Cr, 

per 
cent. 

1 

0.55 

0.41 

0.22 

0.03 

0.02 

1.53 

1.14 

Annealed 

75,000 

31 

66 

2 

0.18 

0.27 

0.05 

0.04 

0.02 

1.28 

1.59 

Annealed 

51,000 

37 

71 

3 

0.15 

0.34 

0.13 

0.02 

0.01 

1.28 

0.37 

Annealed 

42,000 

38 

64 

4 

0.25 

0.32 

0.10 

0.03 

0.02 

1.45 

1.20 

Test  piece 

81,500 

35 

68 

5 

0.25 

0.32 

0.10 

0.03 

0.02 

1.45 

1.20 

Full  size  eye 

80,900 

7 

49 

'bar 

6 

0.40 

0.74 

0.24 

0.03 

0.02 

3.45 

1.20 

W830,  A371 

175,000 

10 

43 

7 

0.36 

0.53 

0.11 

0.04 

0.01 

1.53 

0.70 

W830,  A566 

125,000 

20 

65 

8 

0.21 

0.41 

0.22 

0.03 

0.02 

3.52 

1.11 

W830,  A682 

75,000 

24 

66 

9 

0.98 

0.44 

0.16 

0.01 

0.01 

2.02 

0.98 

W843,  A427 

186,000 

10 

46 

10 

0.48 

0.44 

0.16 

0.01 

0.01 

2.02 

0.98 

W893,  A649 

120,000 

18 

61 

11 

0.38 

0.28 

0.27 

0.02 

0.01 

3.01 

0.65 

W843,  A649 

90,000 

25 

69 

474 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Chromium-vanadium  Steels. — These  steels  are  the  latest  development 
and  have  gained  an  extensive  market.  They  are  much  like  chrome-nickel 
steels  but  have  a  greater  contraction  of  area  for  a  given  elastic  limit. 
They  are  much  easier  to  machine;  a  chrome-vanadium  steel  with  an 
elastic  limit  of  150,000  Ib.  may  be  machined  rapidly,  while  a  chrome- nickel 
steel  of  the  same  strength  would  quickly  dull  a  tool  if  cut  at  the  same 
speed. 

Chrome-vanadium  steel  is  more  free  from  surface  imperfections  than 
other  steels  containing  nickel,  vanadium  improving  the  quality,  and 
though  vanadium  is  much  more  costly  than  nickel,  the  smaller  amount 
required  enables  chrome-vanadium  steel  to  compete  with  nickel-steel 
regarding  cost. 

Chrome- vanadium  steel  is  nearly  always  used  in  a  heat-treated  condi- 
tion, although  there  are  some  exceptions.  The  properties  of  chrome- 
vanadium  steel  are  given  in  Table  78. 

TABLE  78. — PROPERTIES  OF  CHROME-VANADIUM  STEEL 


No. 

Composition 

Condition 

SB, 

Elongation 
in     in., 
per  cent. 

Contraction, 
per  cent. 

c, 

per 
cent. 

Mn. 
per 
cent. 

Si, 
per 
cent 

s, 

per 
cent. 

P, 
per 
cent. 

Cr, 

per 
cent. 

V, 

per 
cent 

1 

0.57 

0.84 

0.27 

0.03 

0.01 

1.36 

0.31 

Natural  state 

75,750 

28.1 

68.5 

2 

0.46 

0.48 

0.20 

0.02 

0.01 

1.17 

0.14 

Natural  state 

52,500 

34.0 

71.0 

3 

0.18 

0.32 

0.18 

0.02 

0.01 

0.74 

0.20 

Natural  state 

42,900 

43.0 

75.0 

4 

0.30 

0.65 

0.10 

0.04 

0.04 

0.90 

0.18 

Annealed 

45,000 

35.0 

69.0 

5 

0.30 

0.65 

0.10 

0.04 

0.04 

0.90 

0.18 

W899,  A704 

101,000 

20.0 

64.0 

6 

0.30 

0.65 

0.10 

0.04 

0.04 

0.90 

0.18 

W899,  A454 

180,400 

10.0 

43.0 

7 

0.30 

0.65 

0.10 

0.04 

0.04 

0.90 

0.18 

W899,  A315 

200,000 

10.0 

52.0 

8 

0.28 

0.45 

0.26 

0.02 

0.01 

1.00 

0.18 

O  899,  A676 

79,000 

34.0 

75.0 

9 

0.40 

0.75 

0.26 

0.01 

0.01 

1.00 

0.17 

O  926,  A676 

120,000 

20.0 

53.0 

10 

0.40 

0.75 

0.26 

0.01 

0.01 

1.00 

0.17 

O  926,  A426 

200,000 

11.0 

48.0 

11 
12 
13 

0.57 
1.06 
0.41 

0.37 
0.36 
0.49 

0.20 
0.22 
0.12 

0.02 
0.02 
0.03 

0.01 
0.02 
0.03 

0.69 
0.95 
1.09 

0.22 
0.11 
0.11 

A426 
A648 

177,500 
126,750 
77,250 

14.0 
21.0 
33.0 

57.0 
49.0 
70.0 

A754 

14 

0.25 

0.50 

0.10 

0.03 

0.02 

0.95 

0.75 

113,100 

18.0 

56.0 

Samples  11,  12  and  13  were  hardened  before  being  drawn  at  the 
temperature  given. 

The  foregoing  tables  place  an  excellent  variety  of  steels  at  the  dis- 
posal of  the  designer,  enabling  him  to  meet  every  demand. 

Heat  treatment  has  little  effect  upon  the  modulus  of  elasticity,  which 
for  all  steels  is  between  28  and  30  million,  the  average  value  being  given 
in  Table  73. 

Aluminum  castings,  alloyed  with  copper,  used  for  automobile  and 


GENERAL  CONSIDERATIONS 


475 


similar  engine  crank  cases,  may  be  taken  as  having  a  tensile  strength 
equal  to  cast  iron. 

The  chart  of  Fig.  326  is  given  to  aid  in  the  selection  of  factors  of  safety 
when  a  fraction  of  the  maximum  load  is  repeated  or  reversed;  a  straight- 
line  variation  is  assumed,  and  as  previously  stated,  the  elastic  limit  is 
taken  as  a  basis  for  wrought  iron,  forged  steel  and  steel  castings,  and  the 
ultimate  strength  for  cast  iron  and  other  brittle  materials. 

In  selecting  working  stresses,  the  smaller  machines  belonging  to  a 
heavy  class  are  made  proportionally  heavier  than  the  larger  machines; 
the  stresses  are  reduced  to  bring  this  about.  But  in  a  small  class  of 


5 

Wro't  Iron  a  Steel  i       1.25      1.5       1.75       i       1Z5     Z.50    2.75       : 

i 
t 

<a 

n        n     •>      »     i        ?.'5       3         3.'5        4        4.5        5        55        € 

Cast  Iron        4        5     -  i 

I         8        9        19        II        1 

!  Loads  of  Same  Sign 

Repeated  Load 

c 
c 

0 

2 

CJ 

ro 
LO 

CO 
CTi 

1 

| 

X. 

Reversed  Load 

75 

C 

s 

s 

|  Loads  of  Opposite  Sign 

[ 

rn 

^s. 

s"~ 

vD 
•CO 

^v 

X 

X 

FIG.  326. 

machines,  such  as  automobile  and  airplane  engines,  the  working  stresses 
are  often  higher  than  in  heavy  rolling-mill  engines.  No  one  would 
think  of  using  a  J^-in.  bolt  in  a  place  where  strength  is  required  in  the 
latter,  even  though  calculation  showed  it  ample,  while  for  the  automobile 
engine  a  J^-in.  bolt  is  a  large  bolt  relatively. 

Machines  to  be  handled  by  unskilled  labor  should  have  a  high  factor 
of  judgment;  they  should  be  as  near  "fool  proof "  as  possible  without 
removing  all  profit. 

160.  Selection  of  Formulas. — In  the  more  simple  applications  of 
load  there  is  little  divergence  of  opinion  regarding  the  formulas  applying 
to  calculations  for  stiffness  and  strength,  but  for  loading  involving 
complicated  straining  actions,  different  formulas  are  sometimes  used 
for  the  same  purpose,  giving  results  often  widely  at  variance.  This  is 
to  a  lack  of  agreement  with  tests  or  to  a  lack  of  test  data  in  the  case 


476  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

of  the  more  theoretical  formulas,  and  in  some  cases  to  the  impossibility 
of  a  satisfactory  rational  treatment,  leading  to  the  use  of  empirical 
formulas.  The  formulas  most  commonly  subjected  to  a  variety  of 
treatment  are  for: 

1.  Cylinder  walls. 

2.  Combined  bending  and  twisting. 

3.  Rectangular  sections  in  torsion. 

4.  Struts. 

The  relative  importance  of  maximum  stress  and  maximum  strain 
(deformation)  is  involved  in  the  first  two. 

Let  S  =  tensile  or  compressive  unit  stress. 
Ss  =  shearing  unit  stress. 
E  =  modulus  of  elasticity  (Young's  mod- 
ulus) . 
Es  =  modulus  of  transverse  elasticity  (in 

shear) . 

e  =  deformation  (extension  or  compres- 
sion) . 

m  =  the  reciprocal  of  Poisson's  ratio. 
All  stresses  are  in  pounds  per  square  inch. 

Let  the  parallelepiped  in  Fig.  327  be 
acted  upon  by  three  forces,  producing  stresses,  Si,  $2  and  $3-  It  is 
shown  in  any  good  treatise  on  mechanics  of  materials  or  applied 
mechanics  that  the  stress  Si  produces  a  deformation  in  the  direction  of 
action  equal  to: 

Si 
61  =  E 

The  stresses  S2  and  $3  each  produce  a  deformation  in  a  plane  at  right 
angles  to  their  direction  of  action  equal  to: 

^2        1  j  &3        1 

ei  =  W'm  ei  =  E'm 

The  fraction  1/m  is  known  as  Poisson's  ratio  and  is  usually  between  % 
and  y±. 

If  82  or  $3  is  a  compressive  stress,  elongation  occurs;  if  tensile,  the 
result  is  compression  in  the  direction  of  stress  Si.  The  total  deformation 
in  this  direction  is  the  sum  of  these,  or  the  resultant  deformation  is: 

$1   ,     Sz    .    83 
E      mE       mE 

Arranging  so  that  compressive  stress  may  be  given  the  minus  sign,  and 


GENERAL  CONSIDERATIONS  477 

letting  SR  be  the  resultant  simple  stress  which  would  produce  the  elonga- 
tion eR,  gives: 

SR  =  EeR  =  8,-  ^±-^  (5) 

m 

The  same  treatment  may  be  made  in  the  direction  of  $2  or  $3,  but  it  is 
assumed  that  Si  is  the  greatest. 

If  S2  and  $3  are  compressive  and  Si  tensile,  or  vice  versa,  it  is  obvious 
that  SR  is  greater  than  Si.  The  "maximum  strain  theory"  limits  SR 
to  the  safe  tensile  or  compressive  stress.  In  some  applications  S$  is 
zero,  the  equation  becoming: 

SK  =  EeR  =  Sl  -  ^  (6) 

According  to  (6),  if  Si  and  S2  are  both  tension,  SR  is  less  than  Si,  in- 
dicating an  increase  in  tensile  strength  due  to  the  effect  of  S2.  Prof. 
Cotterill  says:  "An  addition  to  the  tenacity  of  a  material,  consequent  on 
the  application  of  a  lateral  tension,  can,  however,  hardly  be  considered  as 
intrinsically  probable,  and  such  direct  experimental  evidence  as  exists  is 
against  the  supposition."  However,  other  prominent  authorities  use 
the  maximum  strain  theory  in  its  entirety,  the  equivalent  stresses  being 
called  "true  stresses"  by  Merriman. 

Wherever  applicable,  the  maximum  strain  theory  will  be  used  in  this 
book  but  only  when  Si  is  of  different  sign  from  S2  and  $3  (tensile  if  the 
others  are  compression  and  vice  versa);  otherwise  the  effect  of  these 
secondary  stresses  will  be  neglected,  insuring  results  always  on  the  safe 
side. 

If,  in  (6),  $2  is  equal  to  Si,  but  is  of  opposite  sign,  it  may  be  shown 
that  a  shearing  stress  Ss  of  equal  intensity  is  produced.  Then  letting 
Si  =  Ss  and  S2  =  —  SS)  (6)  becomes: 


or: 


In  other  words,  a  given  shear  stress  is  accompanied  by  an  equivalent 
direct  stress  of  greater  intensity  than  the  shear  stress.  This  indicates 
that  if  the  maximum  strain  theory  is  correct  for  stresses  of  opposite  sign, 
the  ratio  of  the  resistance  to  shearing  to  that  of  tension  or  compression 
is  equal  to: 

m 
m  +  l 


478  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


If  SR  and  Ss  are  known,  (7)  gives : 


m  = 


1 


SR   —    Sg 


Some  values  of  m  are  given  in  Table  79. 

TABLE  79 


m 

Material 

Goodman 

Johnson 

Average 

Cast  iron       

3  0  to  4  7 

3   8 

Wrought  iron 

3  6 

3  6 

Steel         

3  6  to  4  6 

3  72 

3  8 

Brass 

3  1  to  3  3 

3  06 

3  2 

Copper  .  . 

2  .  9  to  3  0 

3  06 

3  0 

A  safe  value  of  m  for  most  of  the  materials  used  in  heat  engine  construc- 
tion is  3^,  and  this  will  be  used  in  this  book.     This  gives: 


=  fS  =  0.77S* 


(8) 


a  result  agreeing  well  with  values  commonly  given  for  ductile  materials, 
although  some  authorities  claim  that  the  ratio  SS/SR  is  much  less. 

Hereafter  when  SR  is  the  only  stress  appearing  in  the  working  formula, 
the  subscript  will  be  omitted,  S  denoting  the  maximum  allowable,  or  the 
actual  tensile  or  compressive  stress. 

If  Es  is  the  modulus  of  transverse  elasticity,  or  coefficient  of  rigidity, 
it  may  also  be  shown  that: 


And  for  m  = 


2(m  +  1) 
E 


(9) 
(10) 


161.  Cylinder  Walls. — There  are  several  formulas  for  determining  the 
thickness  of  a  cylinder  wall,  some  of  which  were  compared  by  Alfred 
Petterson  in  the  American  Machinist  of  Feb.  15,  1900.  The  one  most 
generally  accepted,  and  probably  the  most  accurate,  is  that  of  Lame, 
which  will  be  given  here.  Let: 


GENERAL  CONSIDERATIONS 


479 


p  =  internal  pressure  in  pounds  per  square  inch. 
ST  =  tangential,  or  hoop  stress  at  inner  surface,  as  given  by  the  original 

formula  of  Lame. 
SP  =  radial  stress  at  the  inner  surface,  due  directly  to  the  pressure  and 

equal  to  it. 

S  =  equivalent  simple  tensile  stress  due  to  ST  and  SP. 
m  =  the  reciprocal  of  Poisson's  ratio. 
r  =  internal  radius  of  cylinder  in  inches. 
TI  =  external  radius  of  cylinder  in  inches. 
D  =  internal  diameter  in  inches. 
t  =  thickness  of  cylinder  wall  in  inches. 
Let: 

7*1         T  + 


±-.1+r< 


The  original  Lame  formula  gives : 

ri2  +  r>. 


n*+l 


(ID 


where  ST  is  obviously  a  tensile  stress. 
The  radial,  or  normal  stress  is  compressive,  and  is: 

n2  -  1 

From  Formula  (6),  Par.  2,  the  equivalent  simple  tensile  stress  is: 


(12) 


+ 


S  =  ST  -  -F = 


m 


+ 


- 


m 


(13) 


FIG.  328. 

Formula  (13),  known  as  Birnie's  formula,  assumes  no  longitudinal 
force  due  to  reaction  of  cylinder  heads,  the  condition  being  shown  by 
Fig.  328,  in  which  the  heads  are  supported  independently  of  the  cylinder. 
This  condition  is  found  only  in  engines  fitted  with  separate  cylinder 
liners,  but  as  longitudinal  stress,  being  tensile,  supposes  a  reduction  of  S, 
or  a  greater  allowable  working  pressure,  it  is  neglected,  as  stated  in  Par. 


480 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


160.     If  longitudinal   stress  is  considered,    the   formula  is   known  as 
Claverino's  formula.* 

Taking  m  as  3^,  (13)  becomes: 

1.3n2  +  0.7 


and 


P 


n2  -  1 

rc2—  1 

1.3n2  +  0.7* 


S  +  0.7p 
=  \S  -  1.3p 

=  n  —  r  =  r(n  —  1)  = 


(14) 
(15) 

(16) 
(17) 


r 

V 

j 

V 

\ 

\ 

\ 

\ 

\ 

X^ 

0            .025            .0 

FIG 

5            .0" 
± 

329. 

5 

.IZ 

Formula  (16)  may  be  written: 


n  = 


The  chart  in  Fig.  329  will  facilitate 
calculation.     If  S  and  p  are  known, 
(n— 1)/2  may  be  found  and  used  in 
(17) ;  or  if  t  and  D  are  known : 
n  -  I         t_ 
2       =  D 

and  from  the  chart  S/p  may  be  found. 
When  the  pressure  is  small  relative 
to  the  working  stress,  a  simple  for- 
mula, such  as  is  used  for  steam  boilers, 
will  give  results  which  are  usually  con- 
IZS  sidered  sufficiently  accurate. 

Assuming  a  pressure  of  150  Ib.  per 
sq.  in.  gage,  an  allowable  stress  of 
1500  Ib.,  a  common  condition  for  stream  engine  cylinders,  the  thin  cyl- 
inder formula  is  in  error  9  per  cent,  as  compard  with  (17).  If  radial 
stress  is  considered  in  the  thin  cylinder  formula  the  error  is  still  6  per  cent. 
Should  the  pressure  be  500  Ib.  and  the  stress  3500,  as  for  a  Diesel  oil  en- 
gine, the  error  is  nearly  12  per  cent.,  and  with  radial  stress,  nearly  8  per  cent. 

*  In  hydraulic  cylinders  for  high  pressure,  Claverino  's  formula  gives  results  more 
nearly  agreeing  with  practice  than  Birnie's  formula.     The  formula  for  n  is: 


+  0.4  p., 
— r^r-  nm 


3H«     Tnen  t  may  be  found  from  (17). 


GENERAL  CONSIDERATIONS  481 

162.  Combined  Bending  and  Twisting.  —  Let: 

S    =  simple  tensile  or  compressive  stress  produced  by  bending,  or 

more  generally,  by  any  simple  application  of  load. 
Ss  =  simple  shearing  stress  produced  by  twisting  or  by  a  simple  shear- 

ing load. 

SR  =  The  equivalent  simple  tensile  or  compressive  stress   which 
would  produce  the  same  deformation  as  that  resulting  from  the 
joint  action  of  S  and  83.     This  is  taken  as  the  actual  stress 
under  a  given  load,  or  maximum  allowable  working  stress. 
M    =  the  bending  moment  in  inch-pounds. 
Ms  =  the  twisting  moment  in  inch-pounds. 
MR  =  a  bending  moment  which,  with  a  given  section  modulus,  would 

produce  SR. 

z  =  modulus  of  section  in  bending. 
zs  =  modulus  of  section  in  torsion. 
m    =  the  reciprocal  of  Poisson's  ratio. 

The  maximum  direct  stress,  tensile  or  compressive,  produced  by  the 
combination  of  a  simple  direct  stress  with  a  simple  shearing  stress  is 
given  by  the  equation  : 


This  is  called  the  major  principle  stress  and  is  sometimes  the  only  one 
considered  ;  but  for  the  maximum  strain  theory,  the  minor  principle  stress, 
of  opposite  sign  and  normal  to  S\,  must  be  considered,  and  is  given  by  the 
equation  : 


Then  from  (6)  ,  Par.  160,  the  equivalent  simple  stress  is  : 


and  when  m  =  3~: 
o 

SR  =  0.35S  +  0.65  VS2  +  W  (20) 

Formulas  (19)  and  (20)  are  general  for  any  case  where  S  and  Sa  may 

be  found  for  the  same  spot,  the  location  giving  the  maximum  value  of 

SR  being  the  weakest  place  in  any  machine  part. 

For  circular  sections  only,  the  surface  shearing  stress  due  to  torsion 

is  uniform,  and  a  maximum  at  all  points  for  a  given  twisting  moment, 

and  is  : 

„         Ms 

Os   =    —  — 
Zs 
31 


482  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  maximum  bending  stress  is  also  at  the  surface  and  is: 


For  all  sections  formed  by  concentric  circles: 

ZS   =  22 

Then: 


Substituting  these  values  of  S  and  Ss  in  (19)  gives: 


(21) 


and  form  =  3-: 

MB  =  0.35M  +  0.65VM2  +  Ms2  (22) 

To  reduce  the  use  of  large  numerical  quantities,  it  is  convenient  to 
take: 


88      k 


, 
_:=fc     then:    -    = 


(23) 


M 

Then  (20)  becomes: 

SR  =  £[0.35  +  0.65 
And  (22)  becomes: 

MR  =  M[0.35  +  0.65 
The  chart  in  Fig.  330  will  aid  in  the 
use  of  (23)  and  (24). 

The  difference  between  the  use  of 
(18)  and  (20)  may  be  illustrated  by  an 
example  : 

Find  the  resultant  stress  in  a  shaft 
10  in.  in  diameter  when  the  simple 
stress  due  to  bending  is  5000  Ib.  and 
the    ratio    of    twisting    to    bending 
moment  is  unity.    By  (18),  Si  =  6035, 
and  by  (20),  SR  =  6350,   the  former 
involving  an  error  of  nearly  5  per  cent. 
if  the  elongation  theory  is  correct. 
One  other  theory  has  received  some  consideration  and  has  been  in- 
corporated  in   a   number   of   text   books.     It   assumes   the   resultant 
shearing  stress  as  the  limiting  stress,  the  resultant  bending  stress  being 
ignored.     It  therefore  naturally  reduces  to  a  twisting  moment  formula 


L5A 
6B 


FIG.  330. 


GENERAL  CONSIDERATIONS  483 

equated  to  the  modulus  of  section  for  torsion.  To  avoid  confusion,  the 
formula  will  not  be  given,  but  the  resultant  shearing  stress  for  the  ex- 
ample just  given  is  SSR  =  3535,  which  indicates  that  the  shaft  might 
safely  be  smaller.  If  the  maximum  strain  theory  is  assumed,  a  corre- 
sponding direct  stress  given  by  (7)  is:  SR  —  4600;  this,  when  compared 
with  resultant  stress  6350  given  by  (20),  is  in  error  27  per  cent.  The 
error  increases  with  small  values  of  k,  and  unless  increasing  factors  of 
safety  are  employed,  undue  tensile  stress  may  be  produced.  This 
"maximum  shear  theory"  is  sometimes  known  as  Guest's  Law,  although 
not  resulting  in  Guest's  formula,  which  is  empirical. 
Letting  the  bending  stress  S  equal  zero  in  (19)  gives: 

m  +  l    „ 
SR  =    ~^T'Ss 

which  is  the  same  as  given  by  (7),  Par.  160.     If  the  bending  moment  is 
zero,  (21)  may  be  written: 

77? 

—-^  .  Sa  X  2z  =  Sszs  =-  Ms  (25) 

which  is  the  equation  for  simple  twisting;  or  if  the  twisting  moment  is 
zero,  MR  =  M.  It  therefore  appeals  that  general  equations  (19)  and 
(21)  give  a  satisfactory  solution  of  combined  bending  and  twisting  for 
most  problems  occurring  in  heat  engine  design,  and  in  their  simplified 
form  given  by  (23)  and  (24)  are  not  difficult  to  apply. 
From  (7) : 


m          tis 
Then: 

m-  1  SR_ 

2m  2SS 

and 

,     _i_     -1  CY 

m  -\-  \  _  Pa 

2m         2SS ' 

Values  of  SR/S8  are  given  by  some  experimenters,  differing  considerably 
from  those  given  here,  and  no  attempt  is  made  to  relate  them  to  m. 
If  desired,  these  values  may  be  substituted  in  (21),  giving  safer  results, 
especially  for  large  values  of  Ma/M.  If  m  =  1,  (20)  is  the  same  as 
Guest's  formula,  although  the  latter  is  not  derived  in  this  way.  This  is 
the  safest  of  all  formulas  for  combined  bending  and  twisting  moment  of 
round  shafts,  but  perhaps  is  unnecessarily  so.  Recent  values  of  SE/Sss 
for  various  grades  of  steel  range  from  1.45  to  2,  a  value  for  mild  carbon 
steel  being  1.546.  This  would  change  (24)  to: 
MR  =  M[0.23  + 


484  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Assuming  the  theoretical  relation  given  by  (7),  this  gives: 

m  =  1.84. 

That  there  is  difference  of  opinion  concerning  combined  bending  and 
torsion  is  well  known,  but  until  the  relation  between  direct  and  shearing 
stresses  is  better  known,  greater  refinement  seems  unwarranted. 

Since  the  above  was  written,  an  interesting  article  on  combined 
stresses  by  Prof.  A.  Lewis  Jenkins  has  appeared  in  the  August  number 
of  the  Journal  of  the  A.S.M.E.,  p.  694;  it  is  also  in  the  Transactions, 
vol.  39,  p.  929. 

163.  Rectangular   Section  in   Torsion. — The   torsional   modulus  of 
section  which  applies  to  circular  sections  does  not  express  the  stress 
relations  in  a  rectangular  section.     The  stress  in  the  extreme  fiber  of  the 
latter,  instead  of  being  a  maximum,  is  zero,  the  maximum  stress  being 
at  the  surface  of  the  center  of  the  long  side.    It  may  probably  be  assumed 
that  the  stress  at  the  center  of  the  short  side  bears  the  same  ratio  to  that 
of  the  long  side  as  their  distances  from  the  center  of  the  section,  inversely. 
These  stresses  reduce  to  zero  at  the  corners  in  a  manner  represented 
by  a  curved  line,  the  exact  form  of  wjiich  is  probably  unknown.     The  best 
expression  for  the  modulus  of  section  of  a  rectangular  section  is  an  empiri- 
cal formula  given  by  St.  Venant  as  follows: 
b  =  the  length  of  the  short  side. 
h  =  the  length  of  the  long  side. 
Ss  =  maximum  shearing  stress,  at  center  of  long  side. 

SBI  =  shearing  stress  at  center  of  short  side. 

Ms  =  the  twisting  moment. 
zs  =  modulus  of  section  in  torsion. 

'     -„&»  «> 

Let: 

6  = 

then: 

Ms  =  z8Ss  =  3  _^\  8x  •  h*Ss  =  ~  -  Sa  (27) 

where 

c  =  3  +  1.8s 

x2 
Then: 

rr  fi  M-S  (f)'>L\ 

SS  =  C-^  (28) 


GENERAL  CONSIDERATIONS 


485 


where  Ss  is  the  maximum  stress,  which  is  at  the  center  of  the  long  side, 
as  just  stated : 

Then  the  stress  at  the  center  of  the  surface  of  the  short  side  may  be 
taken  as: 

o         b  Q        r      MX 

o.si  =  -r  os  =  Cz  • 


Let: 
Then: 


Cx 


/l3 
3  +  l:8s 


(29) 


\ 


\ 


The  chart  in  Fig.  331  may  be  used  to  find  C  and 

164.  Struts.— Two  of  the  most 
important  engine  parts — the  piston 
rod  and  connecting  rod — are  struts. 
Therefore  the  selection  of  a  suitable  . 
strut  formula,  which  shall  be  safe 
while  not  demanding  an  excess  of 
material,  is  essential  to  rational  de- 
sign. As  already  stated,  this  need 
not  be  a  rational  formula;  this  re- 
lieves an  embarrassing  situation,  as 
the  only  formula  which  rests  on  any 
satisfactory  rational  basis  is  the  long 
column  formula  of  Euler,  which,  with 
few  exceptions,  is  outside  the  range 
of  ratios  of  length  to  radius  of  gyra- 
tion found  in  practice. 

Although  this  formula,  for  prac- 
tical  conditions,  is  inferior  to  any 
one  of  several  empirical  formulas,  it 
is  surprising  to  note  that  it  is  still  used  by  some  authorities.     How- 
.ever,  for  exceedingly  long  struts  which  fail  by  buckling,  and  in  which 
the  direct  unit  load  is  small,  Euler's  formula  gives  results  agreeing  well 
with  experiment;  and  as  it  serves  to  fix  a  limit  for  an  empirical  formula 
to  be  discussed  later,  it  is  given  here. 
P  =  ultimate  load  on  strut. 
p  =  ultimate  unit  load  (pounds  per  square  inch) . 

A  =  area  in  square  inches  of  section  of  maximum  stress.     This  is  at  the 
center  when  both  end  conditions  are  the  same. 


.3      A 


FIG.  331. 


486 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


I  =  actual  length  of  strut  in  inches. 

li  =  the  distance  between  two  consecutive  points  of  contrary  flexure, 
sometimes  called  the  effective  length.  It  is  this  length  which  deter- 
mines the  strength  of  the  strut. 

r  =  the  least  radius  of  gyration,  in  inches,  of  section  A. 
E  =  The  modulus  of  elasticity  of  the  material  (Young's  modulus). 
n  =  a  factor  depending  upon  the  condition  of  the  ends  of  the  strut. 
Then: 

=  P  _   ir*E         Tr2nE 

(30) 


From  (30)  it  may  be  seen  that  n  is  used  so  that  the  formula  may  be  given 
in  terms  of  actual  length;  then: 

ll\2 

n  --  (rj 

Table  80  gives  the  theoretical  conditions  usually  considered  in  strut 
discussions.  They  assume  accurate  dimensions,  homogeneity  of  material, 
and  the  application  of  load  at  the  exact  gravity  axis  of  the  strut.  The 
actual  conditions  are  never  accurately  known  in  practice,  and  intermediate 
values  of  n  are  sometimes  used  which  give  values  agreeing  approximately 
with  tests  with  such  end  conditions  as  are  practically  attainable,  and  will 
be  mentioned  later. 

Of  the  several  strut  formulas  in  common  use,  the  most  satisfactory, 

in  the  author's  opinion,  is  an  em- 
pirical formula  given  by  the  late 
Prof.  J.  B.  Johnson  (Modern  Framed 
Structures,  and  Materials  of  Con- 
struction), a  brief  discussion  of  which 
will  now  be  given.  The  formula  is: 

P  =  K-q(±Y  (31) 

_^  which  is  the  equation  of  the  para- 
bola, K  and  q  being  constants. 
Prof.  Johnson  assumed  that: 

K  =  SE 

(see  Par.  159,  Law  8),  and  q  is  to  be  determined  so  as  to  bring  a  parabola 
tangent  to  Euler's  curve.  This  of  course  occurs  when  the  slope  of  the  two 
curves  is  the  same. 

For  convenience  let  p  =  y,  and  1/r  =  x  (see  Fig.  332). 


FIG.  332. 


GENERAL  CONSIDERATIONS 
TABLE  80 


487 


Case 

I, 

» 

End  conditions 

1 

i 

2Z 

H 

One  end  fixed;  the  other  end 
free  and  unguided. 

IV 

2 

1 

i 

i 

Both  ends  pivoted. 

* 

* 

3 

I 

0.7Z 

2 
(nearly) 

One     end     fixed;     the     other 
pivoted,   but    guided   in   the 
direction  of  axis  of  fixed  end. 

4 

1 

0.5Z 

Both  ends  fixed,  with  common 
axis. 

488  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Then  Euler's  formula  becomes: 


And  Johnson's  is: 

y  =  SEqx2. 
The  slope  of  each  curve  is: 

T-I  i         dy  2ir2nE 

Euler,    ~  = 

dx  x* 

Johnson,    ~.=  —  2qx. 

When  the  curves  are  tangent  to  each  other,  y  and  dy/dx  are  the  same  for 
both;  then: 

7r^-  =  SE  -  qx2  (32) 

and: 

••'¥-*'  ..-;      (33) 

Dividing  (32)  by  (33)  gives: 

8, 

x  = x 

qx 

or: 

:*-* 

From  which: 

Substituting  (34)  in  (32)  gives: 

SE-  ^ 
From  which: 

9  =  iS  <35) 

Johnson's  formula  then  becomes: 

.Sir,2  /    7     \     2 

(36) 


The  limit  of  application  of  this  formula  may  be  found  from  (34), 
which  gives: 


(37) 


2q 
or: 


GENERAL  CONSIDERATIONS  489 

For  any  greater  value  of  l/r,  Euler's  formula  is  to  be  used,  which  is: 

'jjjf^^^S;  p  =  !^  (38) 

The  value  of  q  may  be  found  which  gives  close  agreement  with  tests 
under  certain  practical  conditions  and  n  may  be  found  from  (35).  This 
value  of  n  may  be  assumed  to  be  the  same  for  different  materials  tested 
with  the  same  end  conditions.  The  ordinary  test  conditions  are  for: 

1.  Round  ends. 

2.  Pin  ends.     When  the  pins  are  of  substantial  diameter,    friction 
resists  buckling,  making  the  strut  stronger  than  for  a  round  or  pivot- 
ended  strut. 

3.  Flat  ends,  which  are  stronger  than  pin  ends,  but  not  as  strong  as 
fixed  ends. 

For  the  purpose  of  this  book  the  following  values  of  n  will  be  assumed, 
although  they  are  some  smaller  than  those  used  by  Johnson: 

1.  Round  ends,  n  =  0.90. 

2.  Pin  ends,  n  =  1.45. 

3.  Flat  ends,  n  =  2.00.  ;  . 

That  they  agree  with  tests  in  some  cases  at  least  may  be  seen  from  Fig. 
333. 

These  actual  end  conditions  do  not  occur  in  engine  design,  but  are 
given  here  to  assist  the  designer  in  determining  a  value  of  n  which  most 
nearly  represents  his  practical  problem.  Special  cases  may  arise  in  which 
the  theoretical  values  of  n  in  Table  75  may  be  safely  used. 

Figure  333  is  an  ultimate  unit  load  curve  for  four  formulas,  with 
round  ends,  with  the  values  of  n  just  given.  Curve  A  represents  Euler's 
long  column  formula,  and  B,  Johnson's  parabolic  formula.  These  curves 
are  tangent  to  each  other  and  together  form  the  adopted  curve.  Curve 
C  represents  Ritter's  formula  as  given  by  Goodman  (Mechanics  Applied 
to  Engineering),  in  which  the  limiting  stress  at  failure  when  l/r  is  zero 
is  taken  as  the  "crushing  strength  of  a  short  specimen,"  or  practically,  the 
ultimate  tensile  strength.  Curve  D  is  also  Ritter's  formula  as  given  by 
Kimball  and  Barr  (Machine  Design),  in  which  the  limiting  stress  at  failure 
is  taken  as  the  elastic  limit.  With  these  curves  is  plotted,  values  from 
tables  in  the  Pencoyd  handbook  (curve  E)  which  are  the  average  of  many 
tests  of  struts  composed  of  medium  steel  structural  shapes.  To  obtain  a 
fair  comparison,  the  physical  constants  used  in  the  formulas  are  the  same 
as  for  the  material  used  for  the  Pencoyd  tests.  From  a  study  of  the  curves 
it  may  be  observed: 


490 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


1.  That  Euler's  formula  has  little  application  for  values    of   l/r 
ordinarily  found  in  practice. 

2.  That  Hitter's  formula  (Goodman)  gives  results  considerably  above 
the  tests  when  S  is  the  ultimate  tensile  strength. 

3.  That  Hitter's  formula  (Kimball  and  Barr)  gives  values  far  below 
the  test  values  when  S  equals  the  elastic  limit.     It  is  safe,  but  safety  is 
not  the  only  important  consideration  in  good  design. 


70000 


4.  That  Johnson's  formula  follows  the  tests  very  closely  excepting 
for  small  values  of  l/r. 

Ritter's  formula  is  often  given  in  text-books  on  machine  design  and  it 
is  no  doubt  a  good  safe  formula  for  general  design.  It  is  generally  men- 
tioned as  a  rational  formula,  but  as  it  gives  a  higher  stress  for  a  given 
unit  load  when  the  elastic  limit  is  higher,  this  can  not  be  true.  In  a 
certain  gasoline  engine  connecting  rod  the  unit  load  is  7100  Ib.  Ac- 
cording to  Ritter's  formula  the  stress  produced  in  the  rod  is  10,850 
ft>.  when  the  elastic  limit  is  38,000;  but  when  the  elastic  limit  is  120,000, 


GENERAL  CONSIDERATIONS  491 

the  stress  in  the  rod  as  given  by  the  formula  is  18,850,  being  74  per 
cent,  greater.     This  of  course  is  inconsistent. 

It  is  generally  conceded  that  long  struts  fail  when  or  before  the  elastic 
limit  of  the  material  is  reached.  If  Johnson's  parabola  is  made  tangent 
to  Euler's  curve,  it  seems  reasonable  to  assume  that  any  failure  occur- 
ring when  the  unit  load  lies  on  or  below  these  combined  curves  must 
occur  at  or  within  the  elastic  limit.  We  may  then  employ  factors  of 
safety  with  the  elastic  limit  as  a  basis  for  any  value  of  l/r,  which  would 
not  be  feasible  for  formulas  employing  higher  values  for  the  crushing 
strength  of  short  struts,  even  though  correct. 

It  is  probable,  as  stated  in  the  Pencoyd  handbook,  that  higher  ultimate 
loads  would  obtain  for  round  or  square  sections,  due  perhaps  in  part  to 
absence  of  thin,  unsupported  flanges,  and  partly  to  the  ability  to  obtain 
better  end  conditions.  On  the  contrary,  the  elastic  limit  of  the  material 
decreases  as  the  thickness  increases,  and  this  may  more  than  offset 
the  gain  by  symmetry  of  section. 

Johnson's  formula  will  be  employed  in  this  book,  and  using  the  phy- 
sical constants  given  in  Table  73,  the  following  special  formulas  may  be 
derived,  which  are  conservative  for  materials  commonly  employed  in 
engine  construction.  For  cast  iron  there  is  no  marked  elastic  limit. 
The  value  of  K  in  Formula  (31)  is  taken  as  60,000,  this  giving  results 
agreeing  with  tests. 

Round  ends 


Steel.     When  l/r  <  117:    p  =  38,000  -  1.4  ~  (39) 

Wrought  iron.     When  l/r  ^126:    p  =  27,000  -  0.79  (-)'  (40) 

Cast  iron.     When  l/r  ^  59:    p  =  60,000  -  8.5  (-Y  (41) 

Pin  ends 

Steel.     When  l/r  <  147:    p  =  38,000  -  0.87  (-)*  (42) 

Wrought  iron.     When  l/r  ^166:  p  =  27,000  -  0.49^-)  *  (43) 

Flat  ends 

Steel.     When  l/r  <  173:  p  =  38,000  -  0.63  (-Y  (44) 

Wrought  iron.     When  l/r  ^  195:  p  =  27,000  -  0.355  (-)  *  (45) 

Cast  iron.     When  l/r  ^  88:  p  =  60,000  -  0.38  (-\  *  (46) 


492 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


When  a  strut  is  subjected  to  bending  stress,  from  eccentric  loading  or 
otherwise,  the  sum  of  the  bending  stress  at  center  of  strut  and  direct 
unit  load  should  not  exceed  the  unit  load  allowed  by  the  strut  formula, 
using  least  radius  of  gyration.  Should  the  end  conditions  not  be  the 
same  in  all  planes,  as  in  an  engine  connecting  rod,  the  maximum  value 
of  q/r  must  be  used,  as  it  is  obvious  from  (31)  that  this  will  give  the  mini- 
mum value  of  the  ultimate  load. 


40000 


30000 


20000 


10000 


100 


ZOO 


300 


FIG.  334. 

To  facilitate  calculations  of  steel  struts,  the  chart  in  Fig.  334  for 
round,  pin  and  flat  ends  is  given,  being  plotted  from  special  equations 
(39),  (42)  and  (44). 

165.  Clearances  and  Tolerances. — In  the  various  machinists'  hand- 
books and  books  on  general  machine  design  there  are  many  tables  giving 
clearances  and  tolerances  for  different  kinds  of  work  and  also  much  other 


GENERAL  CONSIDERATIONS 


493 


data  relating  specifically  to  the  shop.  While  it  is  necessary  for  the  de- 
signer to  have  such  data,  it  seems  to  be  outside  of  the  scope  of  this 
book,  so  comparatively  little  is  said  on  the  subject.  No  absolute  stand- 
ard has  been  adopted,  but  many  concerns  have  their  own  standards. 
These  should  be  in  the  hands  of  all  designers  in  a  particular  office  so  that 
there  may  be  uniformity  of  practice,  at  least  for  that  office.  The  stand- 
ards should  be  adopted  by  the  cooperation  of  engineering  department 
and  machine  shops,  and  should  not  be  changed  unless  experience  shows 
that  alterations  are  desirable. 

166.  Basis  of  Design. — The  forces  acting  on  engine  parts  are  com- 
monly computed  for  the  maximum  steam  or  gas  pressure  in  the  cylinder. 
If  these  alone  may  be  considered,  it  is  a  simple  matter  to  determine 
stresses  in  both  magnitude  and  kind.  For  large,  slow-speed  engines,  these 


FIG.  335. 

forces,  with  those  produced  by  heavy  weights  such  as  the  flywheel,  are 
all  that  are  practically  necessary  to  consider;  but  for  high-speed  engines, 
the  inertia  forces  may  approach  those  due  to  steam  or  gas  pressure,  and 
may  not  safely  be  overlooked. 

The  forces  acting  normal  to  the  line  of  stroke  are  comparatively  small 
for  most  engine  parts,  so  that  by  the  use  of  combined  indicator  and 
inertia  diagrams  a  comparatively  simple  method  of  selecting  factors  of 
safety  may  be  devised  which  will  cover  ordinary  cases. 

Aside  from  attachment  to  foundation  or  other  supports,  the  frame  is 
not  affected  by  inertia  forces  along  the  line  of  stroke;  indicator  diagrams 
only  are  required  to  determine  the  forces.  This  is  also  true  of  the  cylin- 
der. Practically  all  other  parts  are  affected  by  inertia. 

Single-acting  Engines. — Fig.  335  shows  stroke  diagrams  combined 
with  inertia  diagrams  for  single-acting  steam  and  internal-combustion 
engines.  Reference  may  also  be  made  to  Chap.  XVI.  Whether  the 
resulting  stresses  are  repeated  or  reversed  will  depend  largely  upon  the 
amount  of  inertia.  The  inertia  diagram  may  include  only  the  recipro- 
cating parts  in  determining  the  forces  acting  upon  certain  parts,  or  for 
some  cases  the  connecting  rod  may  be  included. 


494 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  following  formulas  may  assist  in  obtaining  factors.     From  (33) 
and  (34) ,  Chap.  XVI,  if  n  is  the  ratio  of  connecting  rod  to  crank  length : 

z?  i    i 

r  H  _  n  -f-  1 

F^  ~  n  -  1* 
Let: 

P2  FH 


and 


Pi 


=  k. 


The  notation  is  taken  from  Fig.  335.     Then: 

Tl=Pl-FH  =  P,(l  -  k) 
and: 

,    n  —  I 


Let  the  maximum  value  of  T  be  TM'}  then: 


(47) 
(48) 

(49) 


This  may  be  called  the  pressure  factor  and  may  be  a  part  of  the  factor  of 
judgment. 


FIG.  336. 

In  the  single-acting  steam  engine,  compression  may  offset  part  or  all 
of  the  inertia  at  the  head  end  as  shown  in  the  dotted  lines. 

Double-acting  Engines. — In  these  engines,  the  greater  range  of  T  is 
found  in  the  crank-end  diagrams,  and  these  are  shown  for  both  steam- 
and  internal-combustion  engines  in  Fig.  336. 
The  equations  for  this  case  are : 

T,=P,-  Fa  -=  Pi  (l  -  fc  ^-j)  (50) 

and: 

T2  =  P2  +  FH  =  P1(q  +  k)  (51) 

In  the  steam  engine,  compression  may  reduce  P2  and  T2,  making  them 
negative  in  some  cases. 

As  an  aid  in  determining  factors  Table  81  is  given.     As  the  single- 
acting  steam  engine  is  little  used  it  is  omitted. 


GENERAL  CONSIDERATIONS 


495 


I 


(£  (C  (£  &-1  £•< 

000=5 


J 

EM 


•• 


<y«<ucjaj 


O     O 


0 

E1*! 


05000        o        o 


"   " 


II 


o 


.So 

ri 
1! 


3     3 


CJ      O      C      ^      ^  O 

Illll   I 


bfl 

.« 

-P 


.*  »  - 

^ 


- 


bO 


.*  «  - 

S  -e  -g 

ajo8<u 


3    1 


I   -I 


. 


o 


o 


496  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  symbols  TMH  and  TMc  are  maximum  effective  pressures  during 
the  stroke  from  head  to  crank  end,  and  from  crank  to  head  end  re- 
spectively. TMH  may  be  found  from  (47)  or  (48).  It  is  possible  for 
T  to  have  a  maximum  value  at  other  than  dead-center  positions,  espe- 
cially in  steam  engines  when  the  cut-off  is  long,  but  the  difference  will  not 
be  great.  Where  T\  and  772  are  less  than  P1;  the  latter  has  been  used  for 
steam  engine  reciprocating  parts  in  Table  81.  It  is  obvious  that  a  re- 
versal from  one  stress  to  another  becomes  a  repeated  stress  if  one  of  the 
stresses  is  zero. 

The  twisting  load  on  the  shaft  is  practically  a  repeated  load.  The 
load  due  to  weight  of  wheel,  etc.  is  always  a  reversed  load,  and  as  it  is 
difficult  to  use  both  of  these  factors  in  the  combined  stress  formula,  it  is 
probably  better  to  assume  a  reversed  load  when  combining  with  a 
wheel  load,  making  the  factor  of  judgment  unity  except  for  very  severe 
service. 

When  cranks  are  in  a  position  to  receive  a  large  turning  effort — 
perhaps  40  degrees  or  more  away  from  either  dead  center — the  stresses 
due  to  both  bending  and  twisting  may  be  taken  as  repeated  stresses  with 
a  factor  of  safety  of  3  (for  ductile  materials,  and  6  for  brittle  materials) . 
In  this  case,  actual  pressures,  including  inertia,  must  be  used,  the  factor 
not  being  based  upon  maximum  steam  or  gas  pressure,  as  no  pressure 
factor  is  used.  It  is  difficult  to  give  pressure  factors  near  mid-stroke, 
especially  for  the  internal-combustion  engine.  A  factor  of  judgment- 
may  be  used  if  desired. 

In  designing  engines,  it  is  a  simple  matter  to  plot  diagrams  such  as 
are  given  in  Figs.  335  and  336,  and  this  is  recommended. 

As  an  illustration,  application  will  be  made  for  five  values  of  head- 
end inertia  as  follows:  when  FH  =  PI,  or  k  =  1;  when  FH  =  Pi/2,  or 
&  =  0.5;  and  when  k  =  0.25,  0.125  and  zero.  The  condition  when  k  = 
0  would  only  apply  at  starting  and  would  be  used  only  for  engines  work- 
ing with  frequent  stops.  Values  of  n  and  q  are  also  assumed. 

It  may  be  seen  from  Table  81  that  aside  from  the  cylinder,  frame,  and 
the  connecting  rod  for  vibration,  the  standard  factors  for  which  are  obvi- 
ous, there  are  but  two  groups,  the  reciprocating  parts  and  the  revolving 
parts.  Factors  for  these  two  groups  will  be  tabulated  in  Table  82;  the 
several  values  of  inertia  will  be  assumed  as  the  inertia  of  the  parts  in- 
volved; for  parts  acted  upon  by  the  inertia  of  fewer  other  parts,  the  fac- 
tors may  be  reduced  a  small  amount. 

For  obtaining  the  standard  factor  fAt  Fig.  326  may  be  used.  For  cast 
iron  and  other  brittle  materials  the  factor  of  safety  is  based  upon  the 
ultimate  strength,  and  the  values  of  Table  82  should  be  multiplied  by  2, 


GENERAL  CONSIDERATIONS 


497 


or  in  case  of  reversal  the  compressive  stress  may  be  assumed  to  have  one- 
fifth  of  its  actual  value  as  stated  in  Par.  1. 

A  factor  of  judgment  may  be  applied,  or  fT  may  be  considered  as  a 
part  of  the  factor  of  judgment  and  may  in  some  cases  be  equal  to  it. 
It  must  be  remembered  that  the  factor  finally  determined  assumes  the 
part  acted  upon  only  by  the  maximum  steam  or  gas  pressure,  even  though 
in  reality  the  maximum  force  is  not  a  maximum  when  the  pressure  is 
maximum.  This  is  provided  for  by  fT  and  fA. 

TABLE  82 


Single-acting     en- 
gines 

Double-acting  engines 

Parts 

* 

4-cycle  internal- 
combustion  engines 

Steam 

4-cycle  internal- 
combustion 

n  =  4      g  =  0.2 

n  =  5            g  =  0.5 

n=f4                g  =  0.2 

'•• 

/. 

fT 

fo/T 

fa 

fT 

fo/T 

/. 

fT 

Aft 

1.000 

5.4 

1.00 

5.4 

5.34 

.50 

8.00 

5.75 

1.20 

6.90 

Reciprocating  parts  

0.500 

6.0 

0.50 

3.0* 

5.50 

.00 

5.50 

5.15 

0.70 

3.60 

0.250 

4.0 

0.75 

3.0 

6.00 

.00 

6.  Qp* 

5.65 

0.85 

4.80 

0.125 

3.4 

0.87 

3.0 

6.00 

.00 

6.00 

5.85 

0.92 

5.40* 

0.000 

3.0 

1.00 

3.0 

6.00 

.00 

6.00 

6.00 

1.00 

6.00 

1.000 

3.0 

1.00 

3.0 

3.70 

.50 

5.55 

4.00 

1.20 

4.80 

0.500 

6.0 

0.50 

3.0 

5.00 

.00 

5.00* 

6.00 

0.70 

4.20* 

Revolving  parts  

0.250 

4.4 

0.75 

3.3 

5.70 

0.83 

4.75 

4.60 

0.85 

3.90 

0.125 

4.0 

0.87 

3.5* 

5.05 

0.92 

4.65 

4.00 

0.92 

3.70 

0.000 

3.6 

1.00 

3.6 

4.50 

1.00 

4.50 

3.60 

1.00 

3.60 

Table  82  may  be  used  with  judgment  by  calculating  FH  and  getting 
the  value  of  k.  For  Corliss  engines  with  a  single  eccentric,  the  cut-off 
would  never  be  as  long  as  assumed  in  fixing  the  value  of  q. 

It  is  obvious  that  for  large,  slow-speed  engines,  k  would  not  likely  be 
as  great  as  unity,  but  it  may  be  greater  in  high-speed  engines.  It  is 
best  to  lay  out  diagrams  as  previously  suggested,  or  calculate  the  inertia 
at  both  ends  of  the  stroke,  but  Table  82  shows  some  interesting  results 
and  may  be  used  as  a  guide  in  determining  factors.  At  starting,  the 
maximum  gas  pressure  is  applied  to  the  parts  of  internal-combustion 
engines,  and  for  this  the  pressure  factor  is  unity,  and  the  standard  factor 
should  never  be  less  than  for  static  loading.  The  minimum  value  in 
Table  82  amply  provides  for  this. 

A  comparison  of  the  factors  for  the  steam  engine  and  the  single- 
acting  internal-combustion  engine  will  show  why  the  latter,  although 
carrying  higher  pressure,  is  not  proportionally  heavier,  but  sometimes 

32 


498  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

lighter  than  the  steam  engine.  Factors  marked*  may  be  taken  as  safe 
factors  for  ordinary  conditions  when  a  careful  analysis  is  not  to  be  made. 

In  all  design  it  must  be  kept  in  mind  that  cost  of  production  will 
always  be  a  great  factor  in  selection.  As  stated  at  the  beginning  of  Par. 
142,  economy,  effectiveness,  safety  and  durability  are  the  criteria  of  good 
design,  and  to  strike  a  proper  balance  between  these  is  the  work  of  the 
designer.  There  is  no  limit  to  the  training  and  experience  which  may 
be  applied  to  this  work. 

Reference  will  be  made  to  this  paragraph  in  the  treatment  of  the 
various  details. 


CHAPTER  XXII 

CYLINDERS 
Notation. 

t  =  thickness  of  wall  in  inches. 
D  =  diameter  of  cylinder  in  inches. 
Ds  =  diameter  of  steam  inlet  in  inches. 
DE  =  diameter  of  exhaust  outlet  in  inches. 

d  =  diameter  of  piston  rod  in  inches. 
ds  =  diameter  of  stud  in  inches. 
di  =  diameter  of  stud  at  root  of  thread. 
A  =  piston  area  in  sq.  in.  =  irD2/4. 

a  =  area  of  steam  passage  in  square  inches. 

c  =  clearance  distance. 

p  =  unbalanced  pressure  in  pounds  per  square  inch. 
fa  =  standard  factor  of  safety, 
/s  =  factor  of  judgment. 

S  =  tensile  stress  in  pounds  per  square  inch. 
SP  =  piston  speed  in  feet  per  minute. 
V  =  nominal  average  velocity  of  steam  or  gas  in  feet  per  minute, 

167.  The  Cylinder. — The  material  most  used  for  cylinders  is  hard, 
close  grained,  gray  cast  iron  known  as  cylinder  iron.  If  properly  made, 
cast  iron  cylinders  have  sufficient  strength,  and  with  proper  lubrication 
and  care,  the  cylinder  bore  and  valve  seat  attain  a  smooth,  glass-like 
surface  which  gives  but  little  frictional  resistance  and  wears  almost 
indefinitely. 

Strength  calculations  are  simple  and  few,  the  principal  dimension  to 
be  determined  being  the  thickness  of  the  wall.  Cylinder  formulas  were 
discussed  in  Par.  161,  Chap.  XXI,  in  which  Formula  (17)  gives  the  thick- 
ness of  the  wall.  The  chart  in  Fig.  328  of  the  same  chapter  greatly 
facilitates  calculation.  Allowing  for  other  considerations  than  strength, 
(17)  may  be  written 

t  =  ?~-D  +  k  (1) 

where  k  provides  for  possible  reboring,  unequal  thickness  of  wall  due  to 
shifting  of  cores  in  casting,  porous  material  and  other  defects.  This 
constant  may  range  from  0.25  to  0.5,  depending  upon  the  quality  of 

499 


500  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

foundry  work  produced  in  a  particular  foundry.  For  very  small 
cylinders  such  as  for  small  gasoline  engines,  k  may  be  very  much  less; 
and  for  cylinder  linings,  such  as  are  used  in  Diesel  engines,  it  may  be 
zero,  as  when  wear  occurs  the  lining  may  be  replaced. 

The  value  of  n  in  (1)  is  given  by  (16),  Chap.  XXI,  and  is: 


S  +  0.7p 

n 


The  stresses  in  cylinder  walls  are  repeated  stresses  due  to  the  steam  or 
gas  pressure  only,  inertia  having  no  effect.  Table  72  of  Chap.  XXI  gives 
6  as  the  standard  factor  of  safety  fA.  Using  the  tensile  strength  of  cast 
iron  from  Table  74,  Chap.  XXI,  which  is  16,000,  gives  a  maximum  working 
stress  of  2670  Ib.  per  sq.  in.  A  common  stress  for  steam  cylinders  is 
1500;  this  gives  a  factor  of  judgment/3  of  1.78,  which  is  reasonable,  due 
to  the  disastrous  effect  of  steam  cylinder  failure,  caused  by  escaping 
steam. 

Much  higher  stresses  are  used  in  gas  engine  cylinders  —  as  high  as  3500 
Ib.  and  even  higher.  While  pressures  in  such  cylinders  are  probably 
more  erratic  than  in  steam  cylinders,  there  is  no  escaping  steam  in  case 
of  failure,  and  due  to  high  temperature,  failure  is  more  apt  to  be  due  to 
unequal  expansion,  the  stresses  due  to  which  are  increased  by  thick 
cylinder  walls.  Neglecting,  the  factor  of  judgment  but  retaining  the 
standard  factor  6,  a  working  stress  of  3500  Ib.  would  necessitate  an  ulti- 
mate tensile  strength  of  21,000  Ib.;  this  is  not  excessive  for  the  grade  of 
material  which  should  be  used  for  cylinders,  the  value  16,000  being  very 
conservative  and  suitable  for  general  use  when  the  quality  of  the  material 
is  uncertain. 

Internal  stress  due  to  shrinkage  of  the  metal  in  cooling  is  often  much 
greater  than  operative  stresses.  For  this  reason  it  is  often  stated  that 
rational  formulas  do  not  apply  to  cast  iron.  While  this  is  partly  true, 
there  seems  to  be  no  better  method  of  determining  dimensions  if  the 
formulas  are  used  with  judgment. 

Good  foundry  practice  with  careful  cooling  of  castings  is  as  essential 
as  good  design.  The  latter  consists  in  an  even  distribution  of  material, 
making  the  thickness  as  uniform  as  possible,  and  avoiding  pockets 
and  constricted  passages,  the  cores  for  jwhich  are  apt  to  be  displaced  in 
casting. 

Walls  other  than  those  of  the  cylinder  barrel  may  be  made  according 
to  (1),  omitting  k  if  subject  to  the  high-pressure  steam;  if  containing 
exhaust  steam  they  may  be  %  as  thick.  With  small  cylinders  this  may 
lead  to  too  much  difference  in  thickness,  in  which  case  steam  and  exhaust 


CYLINDERS  501 

passages  may  have  walls  of  the  same  thickness.  Formula  (1)  is  only  an 
empirical  formula  when  used  for  other  than  circular  passages. 

Cylinder  flanges  may  be  about  1.2  times  the  thickness  of  the  barrel 
wall. 

Flat  surfaces  should  be  carefully  supported  with  ribs,  but  ribs  should 
not  be  too  generally  used,  especially  on  the  outside  of  the  casting.  It  is 
better  to  have  the  cylinder  barrel  free  from  ribs,  as  a  cracked  rib  may 
mean  a  ruptured  cylinder  later  The  strength  of  flat  surfaces  when 
round  or  rectangular,  and  not  supported  by  ribs,  may  be  checked  by 
flat  plate  formulas.  For  rectangular  plates  with  uniform  load,  Leut- 
wiler's  Machine  Design  gives: 


'  -  <•     -.  (3) 


in  which  a  is  the  length  and  b  the  breadth  of  the  plate  and  K  is  Bach's 
coefficients.  For  plates  supported  at  the  edge,  K  is  0.565;  when  fixed 
at  the  edge  K  is  0.375  when  the  material  is  cast  iron.  For  mild  steel, 
K  is  0.36  and  0.24  for  free  and  fixed  edges  respectively. 

For  circular  plates  with  uniform  load  the  same  authority  gives: 

(4) 

where  a  is  the  diameter.  For  cast  iron,  K  is  0.3  and  0.2  for  free  and  fixed 
edges  respectively,  and  for  mild  steel,  0.19  and  0.13.  For  most  practical 
cases  the  plates  may  be  considered  as  neither  free  or  fixed.  It  is  well  to 
calculate  both  ways  when  a  compromise  may  be  made  with  judgment. 

It  is  well  when  possible  to  provide  for  free  expansion  of  the  cylinder 
barrel.  It  is  true  that  the  usual  Corliss  engine  cylinder  has  the  cylinder 
ends  connected  by  the  steam  chest  and  exhaust  passage,  which,  with 
the  cylinder  itself,  carry  three  different  mean  temperatures;  these 
cylinders  operate  indefinitely  without  failure,  even  with  superheated 
steam  in  some  cases,  and  when  they  do  fail  it  may  usually  be  traced  to 
some  other  cause.  However,  for  high  superheat,  the  free  cylinder  is 
probably  better. 

In  any  steam  cylinder,  the  exhaust  steam  should  not  come  in  contact 
with  walls  having  live  steam  on  the  other  side,  as  heat  from  the  latter  is 
quickly  taken  up  by  the  exhaust  steam,  resulting  in  condensation  and  loss. 
An  air  space  should  always  be  left. 

Several  points  in  cylinder  design  may  be  shown  by  Figs.  337  and  338, 
which  are  for  the  cylinder  drawings  for  the  20  by  48-in.  Corliss  engine  of 
Chap.  XII.  The  covers  of  the  valve  chambers  are  called  bonnets,  the 


502 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


r "/-'•- -H 

K-H - 


FIG.  337. 


FIG.  338. 


CYLINDERS 


503 


front  bonnet  forming  a  bracket  for  the  valve  gear.  The  back  bonnets 
for  the  exhaust  valve  chambers  are  tapped  for  water  relief  valves.  Parts 
of  the  cylinder  depending  upon  the  valve  gear  are  given  in  Chap.  XX. 

Counterbore  allows  for  reboring  and  for  the  piston  ring  to  "  wipe  over" 
a  small  amount  to  prevent  wearing  a  shoulder  in  the  cylinder  near  the 
end  of  the  stroke.  There  is  no  definite  rule  for  the  amount,  but  Jle  in. 
is  usually  ample;  too  great  an  amount  may  result  in  slapping  of  the  ring 
at  the  ends  of  the  stroke. 

The  clearance  c  is  as  small  as  is  considered  safe  and  may  be  as  small 
as  H  in. ;  most  builders  prefer  to  allow  ample  in  the  interest  of  safety  and 
from  Y±  to  M  in-  ^  more  usual,  depending  upon  the  size  of  the  cylinder. 
Table  83  gives  values  of  counterbore  and  clearance  for  steam  cylinders  of 
different  size,  which  may  be  used  as  a  guide.  In  view  of  the  discussion  of 
clearance  volume  in  Par.  46,  Chap.  IX,  and  Par.  62,  Chap.  XII,  the 
larger  and  safer  values  of  c  seem  to  be  justified. 


TABLE 

83 

Cylinder  diameter 

10 

12 

14 

16 

18 

20 

22 

24 

Counterbore  

12H 

20% 

22% 

24% 

Clearance  

K 

y 

H 

V 

Cylinder  diameter 

26 

28 

30 

32 

34 

36 

38 

40 

Counterbore  

26^ 

28  % 

30>^ 

32^ 

34^ 

Clearance       .                        ... 

3/ 
/8 

H 

Y 

For  larger  sizes  the  clearance  and  allowance  for  counterbore  may  be 
taken  the  same  as  the  maximum  values  in  the  table. 

In  internal-combustion  engines  and  the  uniflow  steam  engine,  clear- 
ance is  determined  by  the  required  compression  pressure. 

168.  Inlet  and  exhaust  passages  in  steam  cylinders  are  dependent 
upon  the  allowable  nominal  steam  velocities;  the  area  of  a  passage  is 
given  by  Formula  (2),  Chap.  XX,  and  is: 

SPA  ,,, 

a  =    -y- 

in  which  V  will  be  taken  in  ft.  per  min.  in  this  chapter.     The  values  of  V 
are  given  by  Formulas  (3)  and  (4),  Chap.  XX. 

In  steam  engines  the  steam  and  exhaust  connections  are  for  standard 
round  piping;  then  we  may  write: 


as 


aE 


504 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


where  subscripts  S  and  E  denote  steam  and  exhaust  respectively.     Then 
(5)  becomes: 


and: 


D' = 


(6) 


(7) 


The  next  larger  diameters  of  standard  wrought  pipes  may  be  used  if  Ds 
and  DE  are  intermediate  values.  Values  of  Vs  and  VE  may  be  found  from 
the  curves  of  Fig.  339,  which  were  plotted  from  (3)  and  (4),  Chap.  XX. 


9000 


4000 


40  50 

D 
FIG.  339. 


60 


Formula  (5)  may  also  be  used  for  internal-combustion  engines  by 
using  the  proper  value  of  V.  This  also  applies  to  (6)  and  (7)  when  the 
passages  are  of  circular  section.  Lucke  says  that  V  should  not  exceed 
3000.  Measurements  of  the  cylinder  of  a  well-known  automobile  engine 
gives  for  the  entrance  to  the  cylinder  a  velocity  of  8400  ft.  per  min.  and 
for  the  exit  6800  ft.,  for  a  piston  speed  of  1000  ft.  per  min.  A  modern 
vertical  gas  engine  gives  a  gas  velocity  through  the  inlet  opening  of  the 
cylinder  head  of  5380  ft.  per  min.,  and  through  the  exhaust  opening, 
7850  ft.,  with  a  piston  speed  of  640  ft.  These  velocities  are  all  nominal 
velocities  as  explained  in  Par.  142,  Chap.  XX.  Velocities  through  ports 
and  valves  of  both  steam  and  internal-combustion  engines  are  discussed 
in  Chap.  XX. 

169.  Cylinder  heads  are  largely  of  empirical  design  as  there  are  no 
satisfactory  rational  formulas  for  determining  the  thickness  .  of  metal. 
In  shallow  heads  which  set  into  the  cylinder  but  a  small  distance,  a  flat 
circular  plate  formula  may  be  used  as  a  check.  For  deep  heads,  such  as 
used  on  Corliss  cylinders,  the  strength  depends  largely  upon  the  ribs, 
which  form  sort  of  a  truss.  Such  a  head  is  shown  in  Fig.  340. 

As  a  guide  the  following  dimensions  are  given: 

ti  =  1. It  tz  =  0.6h 


CYLINDERS 


505 


It  will  be  noticed  that  the  head  has  a  snug  fit  in  the  cylinder  for  but  a 
short  distance,  from  %  to  1  in.,  depending  on  the  cylinder  diameter,  after 
which  it  is  cut  away  J-^2  m->  or  sometimes  less.  It  was  once  common  to 
have  the  entire  flange  surface  fit  tight  against  the  cylinder,  with  paper 
or  some  other  packing  between ;  but  now  an  annular  space  only  comes  in 
contact  with  the  cylinder;  this  sur- 
face, about  ^4  in.  wide,  permits  a 
greatly  increased  pressure  per  sq.  in. 
and  is  more  easily  kept  steam-tight. 
This  surface  is  sometimes  ground  in 
place,  but  this  is  not  necessary  for  a 
steam-tight  joint. 

It  is  well  to  keep  the  stud  circle 
as  near  the  inner  edge  as  possible 
without  danger  of  breaking  out ;  good 
results  are  obtained  if  this  distance  is  about  1.2  times  the  stud  diameter, 
and  this  same  distance  may  be  taken  to  the  outer  edge  of  the  head. 
Table  84  gives  the  dimensions  of  high-pressure  and  low-pressure  heads, 

TABLE  84 


FIG.  340. 


High-pressure  heads 

Low-pressure  heads 

D,  in. 

DC, 
in. 

DH, 
in. 

Ds, 
in. 

ds. 

in. 

No. 
studs 

A 

for  p=  100 

D, 
in. 

DC, 
in. 

DH, 

in. 

I>fi, 
in. 

ds, 
in. 

No. 

studs 

s, 

for  p=  100 

10 

10% 

14% 

12% 

% 

8 

3260 

34 

34^ 

40^ 

37H 

1% 

20 

5090 

12 

12% 

16% 

14K 

% 

8 

3370 

36 

36H 

42^ 

39M 

1% 

22 

5180 

14 

14% 

18% 

16K 

% 

10 

3670 

38 

38M 

44^ 

41H 

1% 

22 

5780 

16 

16% 

21% 

18% 

1 

10 

3660 

40 

40^ 

46M 

43M 

1H 

24 

5890 

18 

18% 

23M 

21 

1 

12 

3870 

42 

42H 

48M 

45M 

1% 

24 

6470 

20 

20% 

25K 

23 

1 

14 

4080 

44 

44^ 

50K 

47H 

1% 

26 

6560 

22 

22% 

27K 

25 

1 

16 

4320 

46 

46M 

52^ 

49H 

1% 

28 

6680 

24 

24% 

30 

27% 

1H 

16 

4080 

48 

48« 

54M 

51M 

1% 

30 

6760 

26 

26K 

32 

29% 

1H 

18 

4250 

50 

50^ 

57 

53% 

IK 

30 

6240 

28 

28K 

34 

31% 

i« 

20 

4470 

52 

52K 

59 

55% 

1% 

30 

6740 

30 

•30K 

36H 

33K 

1% 

20 

3980 

54 

54K 

61 

57% 

1% 

32 

6800 

32 

32  Y* 

38K 

35K 

1M 

22 

4110 

56 

56H 

63 

59% 

IH 

32 

7330 

34 

34  YZ 

40K 

37  K 

l>i 

24 

4240 

58 

58H 

65 

61% 

m 

34 

7380 

36 

36K 

42>£ 

39K 

IK 

26 

4390 

60 

60H 

67M 

64 

IK 

34 

6440 

38 

38K 

45 

41% 

1% 

26 

4100 

62 

62% 

69K 

66 

IK 

34 

6870 

40 

40M 

47K 

44 

1H 

24 

4060 

64 

64% 

71M 

68 

IK 

36 

6920 

42 

42M 

49K 

46 

1M 

24 

4470 

66 

66% 

74M 

70H 

1% 

36 

6280 

44 

44  K 

51  K 

48 

1H 

26 

4530 

68 

68% 

76H 

72H 

1% 

36 

6670 

46 

46K 

53  H 

50 

1M 

28 

4610 

70 

70% 

78H 

74M 

1% 

36 

7070 

48 

48K 

56>£ 

52  K 

1% 

28 

4260 

72 

72% 

80% 

76% 

1H 

38 

7080 

50 

50K 

58>£ 

54K 

1% 

28 

4650 

74 

74% 

82% 

78% 

1% 

38 

7490 

76 

76% 

85^ 

81 

1% 

38 

6860 

78 

78% 

87K 

83 

1% 

38 

7220 

80 

80% 

89% 

85 

1% 

38 

7500 

506  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

with  the  number  of  studs,  and  the  stress  at  the  root  of  the  thread  for  a 
pressure  of  100  Ib.  per  sq.  in.  The  stress  for  any  other  pressure  may  be 
easily  found  as  it  is  proportional  to  the  pressure.  The  notation  is  as 
on  Fig.  340. 

For  Corliss  high-pressure  cylinders,  a  depth  of  head  equal  to  the  diam- 
eter of  the  valve  is  satisfactory;  the  writer  has  used,  for  high-pressure 
heads  : 

Depth  =  Q.2D  +  1.75  (8) 

and  for  low-pressure  heads: 

Depth  =  0.17D  +  1.8  (9) 

There  is  no  such  similarity  between  the  designs  of  heads  for  internal- 
combustion  engines,  but  the  same  general  principles  apply.  The  load 
on  cylinder-head  bolts  or  studs  is  practically  a  dead  load,  and  the  stand- 
ard factor  of  safety  could  be  taken  as  2  if  the  stress  due  to  screwing  up  were 
known.  It  is  better,  however,  to  take  the  repeated  load  factor  3,  based 
upon  the  total  steam  pressure  and  divided  by  the  total  bolt  area  at  root 
of  threads,  which  is  the  assumption  in  Table  84;  this  may  be  called  the 
nominal  stress. 

Small  bolts  are  much  more  liable  to  be  damaged  by  a  wrench,  so  the 
factor  of  judgment  may  be: 

/s  =  1  +  V  (10) 

as 

The  factor  of  safety  is  then: 


in  which  ds  is  the  stud  diameter.  The  safe  stress  for  machinery  steel, 
from  Table  73,  Chap.  XXI  is: 

38,000 

/ 

High-grade  steel  is  sometimes  used,  especially  for  internal-combustion 
engines. 

With  cylinder-head  castings  for  internal-combustion  engines,  great 
care  must  be  exercised  in  design  and  in  the  foundry  work.  There  must 
be  an  even  distribution  of  metal  with  no  abrupt  changes  from  thin  to 
thick  metal.  Ribs  must  not  be  placed  so  that  they  make  the  casting 
too  rigid  and  prevent  the  uneven  expansion  due  to  difference  in  tempera- 
ture. Cored  passages  should  be  as  easy  as  possible. 

Stuffing  Box.  —  If  any  special  form  of  metallic  packing  is  to  be  used,  the 
stuffing  box  should  be  designed  accordingly,  but  for  the  numerous  soft 


CYLINDERS 


507 


packings  on  the  market,  Fig.  341  gives  good  results  if  proportioned 
according  to  Tables  85  and  86,  and  the  following  formulas: 

A  =  0.15D*        B  =  2ds        C  =  l.5ds        E  =  DB  +  2Ads 
F  =  E  +  2Ads        K  =  DB  -  d 


FIG.  341. 

TABLE  85 

d,  in. 

D,  in. 

L,  in. 

Jf,  in. 

<sx 

>3Hor<5 

^5 

d+iy2 

d+1% 
d+2 

4K 
5« 

6 

3K 

4 

4K 

TABLE  86 


d,  in. 

ds,  in. 

.     .              No.  studs 

3 

%       - 

2 

4 

1 

2 

5 

DS 

2 

6 

1 

3 

7 

1« 

3 

8 

1H 

3 

9 

1^ 

3 

The  stuffing-box  flange  is  made  thick  to  offset  the  weakening  effect 
of  the  box,  which  cuts  away  the  ribs  at  the  center. 

170.  Cylinder  Lagging. — Steam  cylinders  are  covered  with  about  1J^ 
in.  of  some  heat-resisting  material  in  the  form  of  plaster,  such  as  asbestos 
or  magnesia.  Outside  of  this  is  the  lagging,  usually  of  sheet  steel,  and 
provision  should  be  made  for  fastening  this  to  the  cylinder.  This  may 
be  seen  in  some  of  the  designs  which  follow. 


508 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


A  water  jacket  is  required  for  internal-combustion  engines,  except  for 
very  small  cylinders,  which  may  be  air-cooled.  No  lagging  is  required. 

Besides  the  illustrations  of  the  following  paragraphs,  some  features 
of  design  may  be  found  in  Chaps.  Ill  and  V. 

The  foundry  and  machine  shop  should  be  kept  in  mind  with  a  view  to 
reducing  the  cost  of  production,  and  no  design  should  be  used  which  has 
been  known  to  give  trouble  in  operation. 

DESIGNS  FROM  PRACTICE 

171.  Steam  Engine  Cylinders. — The  Corliss  engine  cylinder  of  Figs.  337 
and  338,  while  not  designed  for  an  actual  engine,  is  substantially  the 


FIG.  342. — Bruce-Macbeth  gas  engine  cylinder. 

design  used  by  the  author  for  several  years  and  shows  the  characteristic 
features  of  Corliss  cylinders.  The  cylinder  foot  is  included;  openings  are 
provided  for  the  vacuum  dashpots,  but  in  some  cases  these  are  secured 
to  the  foundation  independently.  The  cylinder  foot  is  usually  bolted 
to  the  foundation. 


CYLINDERS 


509 


172.  Internal-combustion  Cylinders.—  A  vertical  gas  engine  cylinder 
is  shown  in  Fig.  342.  This  is  the  design  of  the  Bruce-Macbeth  Engine 
Co.,  Cleveland,  Ohio.  With  a  stress  of  3500  Ib.  and  a  gage  pressure  of 
400  Ib.,  Fig.  328  and  Formula  (17)  of  Chap.  XXI  give  a  wall  thickness  of 
practically  %  in.  The  actual  thickness  being  1  in.  makes  k  in  (1)  equal 
to  Y±  in.  The  jacket  wall  is  made  equal  to  one-half  the  cylinder-wall 
thickness. 

With  a  gage  pressure  of  400  Ib.,  the  nominal  stress  in  the  cylinder- 
head  bolts  at  root  of  thread  is  8850  Ib.  If  the  elastic  limit  of  the  bolt 


FIG.  343. — Bruce-Macbeth  cylinder  heads. 

material  is  38,000,  the  factor  of  safety  is  4.3.  Taking  the  repeated-load 
standard  factor  as  3,  gives  a  factor  of  judgment  of  1.43  instead  of  2,  as 
given  by  (10);  the  former  is  no  doubt  ample. 

The  head  for  this  cylinder  is  shown  in  Fig.  343.  The  valves  and  valve 
cages  are  shown  in  Chap.  XX. 

Fig.  344  shows  the  air-cooled  cylinder  of  the  Franklin  Automobile 
Engine  Steel  ribs  are  cast  in  the  cylinder  as  shown.  These  are  sur- 
rounded by  a  jacket  and  air  is  drawn  through  the  passages  between  the 
ribs. 


510 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  cylinder  wall,  measured  from  the  bottom  of  the  ribs  is 
thick.  With  a  gage  pressure  of  350  Ib.  and  no  allowance  for  wear,  etc., 
Formula  (1)  of  this  chapter  gives  a  stress  of  3325  Ib.,  while  a  pressure  of 
400  Ib.  gives  3800  Ib.  With  the  standard  factor  of  6  for  repeated  load 
for  cast  iron,  neglecting  the  factor  of  judgment,  these  stresses  give  nec- 


5ectionA-A 


FIG.  344. — Cylinder  of  Franklin  automobile  engine. 

essary  ultimate  strengths  of  20,000  and  22,800  Ib.  respectively.  The 
latter  is  no  doubt  conservative,  as  the  better  grades  of  gray  iron  have  a 
tensile  strength  as  high  as  30,000  Ib. 

There  are  four  %-in.  bolts  holding  the  cylinder  to  the  frame.  By  the 
S.A.E.  standard,  they  have  24  threads  per  inch.  The  stress  at  root  of 
thread  for  direct  thrust  only  is  9000  Ib.  for  a  maximum  explosion  pres- 
sure of  350-lb.  gage,  and  10,300  for  400  Ib.  pressure.  This  gives  a  factor 


CYLINDERS 


511 


of  4.2  and  3.7  respectively  for  the  conservative  value  of  the  elastic  limit 
given  in  Table  73,  Chap.  XXI.  Both  of  these  are  greater  than  the  stand- 
ard factor  3  for  repeated  load,  giving  factors  of  judgment  of  1.4  and  1.23. 
This  would,  no  doubt,  provide  for  the  side  thrust  due  to  the  connecting 
rod.  The  maximum  pull  on  the  bolts  occurs  when  the  product  PNa 
(Fig.  345)  is  a  maximum;  as  before,  neglecting  the 
stress  due  to  screwing  up,  the  stress  is  due  to  a 
repeated  load.  If  n  is  t  he  ratio  of  the  length  of 
connecting  rod  to  length  of  crank,  (12),  Chap.  XVI, 

gives:        • 

„  .  P  sin  6 

[1  0\  2 


The  pull  on  the  bolts  for  the  arrangement  of  bolts  in 
Fig.  344  is: 


FIG.  345. 


Assuming  the  diagram  of  Fig.  274,  Chap.  XX  ap- 
plies to  this  engine — which  is  not  strictly  true — 
the  values  of  PB  were  calculated  for  all  positions  covering  that  giving 
the  maximum  value;  corresponding  pressures  were  taken  from  the  indi- 
cator diagram  in  the  same  figure,  and  the  maximum  combined  stress 
from  these  is  11,300  Ib.  per  sq.  in.,  and  occurs  at  position  1,  or  very  near 
head-end  dead  center.  This  does  not  include  the  inertia  of  the  rod, 
which  might  have  modified  the  result  somewhat.  From  the  same  dia- 
gram, the  stress  produced  by  maximum  gas  pressure  is  8620  Ib. 

The  stress  due  to  PN  is  transmitted  through  the  rod,  so  inertia  forces 
must  be  included  in  the  value  of  P  used  (this  should  properly  be  the  pA 
of  Fig.  204,  Chap.  XVI,  instead  of  PN) ;  but  in  obtaining  the  direct  load 
on  the  bolts  in  line  of  stroke,  inertia  has  no  effect  and  gas  pressure  only 
must  be  used. 

For  the  value  of  mild  steel  in  Table  73,  Chap.  XXI,  the  maximum 
stress  just  found  gives  a  factor  of  safety  of  3.36,  but  S.A.E.  standard 
bolts  have  an  elastic  limit  of  60,000  Ib.,  giving  a  factor  of  5.3,  or  a  factor 
of  judgment  of  1.77. 

In  a  steam  engine,  the  maximum  steam  pressure  continues  to  a  point 
beyond  that  at  which  PB  is  a  maximum,  so  that  the  stress  in  the  bolts 
(neglecting  initial  stress)  is  greater  than  that  produced  by  direct  steam 
pressure  in  the  line  of  stroke. 

Fig.  347  shows  a  side  elevation  and  plan  of  a  pair  of  cylinders  of  the 


512  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


O  ^ 


FIG.  346. — Sections    of    Sturtevant    cylinder.      FIG.  347. — Sturtevant     airplane     engine 

cylinder. 


FIG    348. — Sturtevant  cylinder  heads. 


CYLINDERS 


513 


Sturtevant  airplane  engine,  without  liners.  Fig.  346  shows  a  section 
without  sleeve,  and  a  section  with  the  sleeve  in  place.  The  cylinders 
are  attached  to  the  crank  case  by  bolts  extending  through  lugs  as  shown 
at  section,  and  through  the  cylinder  head. 


Section  D-D 

FIG.  349. — Sturtevant  cylinder  heads. 

Fig.  348  gives  outside  views  of  two  cylinder  heads  cast  enbloc,  with 
rocker  arm  fulcrums;  the  fulcrums  were  formerly  cast  separately  and 
bolted  to  the  head.  Fig.  349  shows  such  a  head,  which  also  better  gives 
the  detail  design. 


33 


CHAPTER  XXIII 

PISTONS 
Notation. 

D  =  diameter  of  cylinder  bore  in  inches. 

F  =  length  of  piston,  or  piston  face,  in  inches. 

L  =  distance  between  the  centers  of  gravity  of  the  two  semi-circles,  in 

inches.  , 

a  =  area  of  half  circle  in  square  inches. 
r  =  radius  of  cylinder  bore  in  inches. 
7*1  =  radius  of  outside  of  piston  ring  when  free,  assumed  constant  for  an 

eccentric  ring. 

t  =  thickness  of  ring  in  inches,  at  any  section. 
tM  =  maximum  thickness  of  ring — opposite  cut. 
w  =  width  of  ring  in  inches. 
7  =  moment  of  inertia  of  ring  section. 
z  =  modulus  of  section  of  piston  section. 
M  =  bending  moment  at  any  section  of  ring. 
E  =  modulus  of  elasticity. 
S  =  bending  stress  in  piston;  also  at  any  section  of  ring  when  in  cylinder, 

exerting  pressure  pN. 

ks=  stress  developed  in  stretching  ring  over  piston  to  place  it  in  groove. 
SM=  maximum  bending  stress  when  ring  is  in  cylinder  (at  section  of 

thickness  tM). 

p  =  maximum  unbalanced  pressure  per  square  inch  in  cylinder. 
pN  =  normal  pressure  in  pound^  per  square  inch  of  ring  upon  cylinder 

wall. 
W=  total  maximum  pressure  in  pounds,  upon  engine  piston. 

173.  Pistons  are  made  in  several  different  styles,  the  most  common  for 
steam  engine  use  being  the  box  piston,  cast  in  one  piece  as  in  Figs.  361 
and  363,  and  the  built-up,  or  bull-ring  piston  shown  in  Fig.  360.  In  the 
former,  ribs  are  usually  provided  for  strengthening  the  piston;  the  holes 
in  one  side  of  the  piston  necessitated  by  the  core,  are  tapped  and  plugged. 
The  cores  between  the  ribs  are  sometimes  connected,  leaving  holes  in 

514 


PISTONS  515 

the  ribs.  This  type  of  piston  has  given  some  trouble,  due  to  the  forma- 
tion of  cracks  where  the  walls  join  hub  and  rim;  this  may  be  due  to 
poorly  distributed  metal  or  improper  cooling  of  the  casting.  This 
design  is  not  much  used  on  very  large  engines. 

The  bull-ring  piston  is  composed  of  the  spider,  or  body,  the  bull  ring 
(sometimes  called  junk  ring),  the  follower,  and,  in  common  with  the  solid 
piston,  one  or  more  packing  rings,  which  prevent  the  leakage  of  steam  past 
the  piston.  The  stiffening  ribs  contain  bosses  into  which  are  tapped  the 
follower  bolts.  A  circular  rib  connects  the  radial  ribs  and  is  provided 
with  set  screws  and  lock  nuts,  by  means  of  which  the  bull  ring  may  be  ad- 
justed to  take  up  wear.  If  wear  is  excessive  the  cylinder  may  be  rebored 
and  a  new  bull  ring  provided.  Trouble  is  sometimes  experienced  by  the 
breaking  of  follower  bolts,  and  as  this  has  been  attributed  to  expansion, 
long  studs,  tapped  into  the  farther  side  of  the  piston  have  been  used  by 
some  builders,  as  shown  in  Fig.  360;  the  stud  is  thus  allowed  to  stretch 
and  bend  slightly  without  undue  stress. 

The  wearing  surface  of  the  piston  is  sometimes  provided  with  rings  or 
strips  of  Babbitt  metal,  or  some  other  anti-friction  metal;  this  is  more 
common  in  large  pistons.  Cast  iron  upon  cast  iron  usually  gives  the  best 
wear,  but  an  occasional  exception  .arises,  when  a  change  to  a  babbitted 
bull  ring  gives  better  results,  due,  no  doubt,  to  some  defect  in  the  metal  of 
cylinder  or  ring. 

When  a  single  packing  ring  is  used  and  made  in  sections,  or  where  two 
"  snap  rings  "  are  used,  the  bull  ring  is  made  in  one  piece,  but  for  the  usual 
single  ring  it  is  in  two  parts,  as  shown  in  Fig.  360. 

When  lightness  is  required  in  a  large  piston,  perhaps  largely  for  the 
purpose  of  balancing,  as  in  marine  engines,  conical  pistons  are  sometimes 
used,  as  in  Fig.  362.  Sometimes  the  box  piston  is  made  conical. 

Single-acting  steam  engines  and  internal-combustion  engines  have 
trunk  pistons,  which  also  serve  as  crossheads,  examples  of  which  are 
shown  in  Figs.  364 and  365.  Trunk  pistons  should  be  designed  so  that  the 
thrust  due  to  the  angularity  of  the  connecting  rod  may  be  transmitted 
from  cylinder  wall  to  wrist  pin  without  distorting  the  piston;  the  proper 
placing  of  ribs  provides  for  this  in  very  light  pistons,  as  seen  in  Fig.  365. 

The  material  used  in  pistons  is  usually  cast  iron,  but  there  is  no  reason, 
aside  from  expense,  why  the  spider  and  follower  of  a  bull -ring  piston  may 
not  be  steel  castings.  Pistons  for  automobile  engines  are  sometimes  made 
of  an  aluminum  alloy,  but  it  has  probably  not  been  tried  sufficiently  long 
to  insure  success.  Morley,  in  his  Strength  of  Materials,  says  that  under 
repeated  stress,  pure  aluminum  tends  to  "  creep,"  or  gradually  fail. 
Aluminum  also  has  a  high  coefficient  of  expansion — more  than  twice  as 


516 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


great  as  for  iron.  If  enough  aluminum  is  used  to  greatly  lighten  the  pis- 
ton, some  trouble  may  possibly  ensue.  According  to  the  American 
Machinists'  Handbook,  aluminum  becomes  granular  and  easily  broken 
at  about  1000°  F. 

The  working  stress  may  be  taken  as  1200  Ib.  for  cast  iron  pistons,  and 
5000  Ib.  for  steel  castings.  From  Tables  73  and  74,  Chap.  XXI,  this  gives 
a  factor  of  judgment  over  the  standard  reversed  stress  factor  of  1.11  for 
cast  iron,  and  zero  for  steel. 

There  is  no  strictly  rational  method  of  determining  the  stress.  The 
flat  plate  formulas  would  apply  only  to  the  simple  plate  piston,  which  is 






u 

5 

FIG.  350. 

rarely  used.  The  piston  has  been  considered  as  a  beam  supported  at  the 
centers  of  gravity  of  the  two  parts  on  either  side  of  a  diameter,  by  one  of 
the  leading  manufacturers,  with  the  maximum  piston  load  applied  at  the 
center.  Possible  rupture  would  occur  along  a  diameter,  which  would  be 
resisted  by  the  weakest  section.  This  would  be  through  the  cored  holes 
of  the  box  piston,  the  plugs  not  adding  to  the  strength.  This  method 
may  be  used  as  a  check  and  will  be  given.  Fig.  350  is  self-explanatory. 
From  mensuration: 

L  =  D* 
2       12a 
and  as: 


a  = 


I   = 

37T 


PISTONS 


517 


From  the  beam  equation: 


0         WL. 

Sz  =  -  -  >   and  as  W  = 

4  4 


Sz 


12 


(1) 


FIG.  351. 

/Si  may  be  found  for  a  given  design,  or  the  necessary  value  of  z  for  a  given 
stress. 

In  most  cases  the  section  may  be  put  into  a  simple  form  as  in  Fig. 
351,  omitting  the  follower  and  bull  ring  from  the  built-up  piston. 

The  center  of  gravity  may  be  found  by  calculation,  locating  the  neutral 
axis  xx]  the  moment  of  inertia  may  then  be  found  and  divided  by  the 
distance  of  the  extreme  fiber  from  the  neutral  axis,  giving  the  minimum 
value  of  z,  and  this  must  be  used  in  (1). 

For  the  conical  piston,  Fig.  352,  an  empirical  rule,  based  in  part  upon 

a  flat  plate  formula,  is: 

/ — ^ 

(2) 
\  £ 

For  cast  iron,  if  S  =  1200  and  0 

T  =  0. 
For  steel  casting,  if  S  =  5000: 

T  =  0.023  Vp5 


Tl  may  equal  0.5  T. 

Using  the  factor  7.2  for  cast 
iron,  as  mentioned  in  connection 
with  Table  72,  Chap.  XXI,  the 
working  stress  may  be  2220  Ib. 
when  the  ultimate  strength  is 
16,000  Ib. 

The  hub  into  which  the  piston 
rod    is    fitted    will    be    considered 
under  the  subject  of  piston  rods.     Trunk  pistons  will  be  discussed  under 
cross-heads. 

The  length  of  piston  F  must  be  such  as  to  allow  of  sufficient  strength 


518  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

and  to  give  ample  wearing  surface.  For  constructional  reasons,  F  is 
proportionally  larger  in  small  than  in  large  engines.  For  steam  engines, 
an  empirical  formula  giving  satisfactory  results  is: 

F  =  f  +  2"  (3) 

174.  Piston  Rings. — Numerous  varieties  of  piston  rings  are  on  the 
market,  but  the  most  common  is  some  form  of  spring  ring,  so  called 
because  it  is  made  larger  in  diameter  than  the  cylinder  bore,  and  when 
in  place,  exerts  a  pressure  upon  the  cylinder  surface.  In  bull-ring  pistons 
this  pressure  is  augmented  by  springs  placed  between  the  packing  ring 
and  bull  ring  at  intervals,  as  shown  in  Fig.  360.  When  flat  springs  are 
used,  lugs  are  cast  on  the  ring  to  keep  them  in  place.  Such  rings  are 
usually  cut  at  an  angle  of  45  degrees  and  a  "keeper"  provided  to  prevent 
leakage,  as  shown  in  Fig.  353. 


\ 


FIG.  353.  FIG.  354. 

Rings  providing  all  of  their  own  spring  are  called  "snap  rings;" 
these  are  either  made  of  uniform  thickness  or  eccentric,  the  latter  ap- 
proximating a  form  which  will  remain  circular  under  uniform  pressure 
at  any  diameter  which  does  not  strain  the  ring  beyond  the  elastic  limit. 
Two  methods  of  cutting  these  rings  are  shown  in  Fig.  354. 

Some  authorities  claim  that  the  ring  of  uniform  thickness  is  superior 
to  the  eccentric  ring;  they  are  more  common,  possibly  because  cheaper 
to  construct.  The  design  of  the  snap  ring  is  often  empirical,  but  a 
fairly  satisfactory  rational  method  is  not  difficult  and  will  be  given.  The 
eccentric  ring  best  lends  itself  to  such  a  discussion. 

The  ring  must  be  designed  so  that  it  will  not  be  stressed  above  its 
elastic  limit  when  in  place  in  the  cylinder,  or  while  sprung  over  the  edge 
of  the  piston  while  placing  in  the  groove.  The  latter  process  is  done 
but  seldom  and  the  stress  may  be  higher  than  the  working  stress,  which, 
however,  is  constant  and  may  be  high. 

Figure  35.5  shows  the  ring  in  place,  Fig.  356  shows  it  free,  while 


PISTON* 


519 


Fig.  357  shows  it  being  placed  in  the  groove.  A  somewhat  smaller 
opening  would  suffice  for  the  last,  but  the  assumption  made  is  safe  and 
simplifies  calculation. 


FIG.  355. 


FIG.  356. 


In  Fig.  355,  the  force  P  (which  is  the  normal  pressure  per  sq.  in. 
multiplied  by  the  projected  area  in  angle  2<x  upon  which  it  acts)  produces 
a  bending  moment  M  at  x.  Angle  2a  may  be  any  angle,  but  it  is  con- 
venient to  take  it  as  20  degrees,  then  increase  it  by  increments  of 


/  // 

N      \   > 

Nv\^ 

: 

XV 

x'''^                               * 

K  P-- 

1C  >1 

\\ 

1 

\\ 

ll 

\\ 

N\ 

// 

FIG.  357. 


20  degrees  until  the  maximum  thickness  at  180  degrees  is  reached.  In 
each  case  P  acts  midway  between  the  cut  and  the  section  x.  In  this  dis- 
cussion the  radius  will  be  taken  to  the  outside  of  the  ring  instead  of 


520  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

at  the  neutral  axis  of  the  sections,  involving  a  small  error;  the  bent 
beam  theory  is  also  ignored. 
In  Fig.  355: 

P  =  pNw-  2r  sina  =  2pNwr  sin  a  (4) 

Taking  moments  about  x  (any  section)  : 

M  =  Pr  sin  a  =  2pNwr2sin2a  (5) 

Also  at  any  section: 

V  * 

and: 

.    "Z 

From  the  general  equation  for  change  of  curvature  from  free  radius 
n  to  radius  of  cylinder  bore  r,  substituting  M  and  /  from  (3)  and  (4)  : 

1        1        M       2S 

^~^  =  EI  =  Ei  =  C°nstant  (8) 

That  is,  the  same  relation  must  hold  for  every  section. 

Assuming  the  stress  developed  by  placing  the  ring  in  the  groove  (Fig. 
357)  as  kS; 

•     :    \    £  ~  rTfc  =  ¥  =  constant        :      y     (9) 

Taking  S  and  t  at  the  maximum  section  (opposite  the  cut)  and  equating 
values  of  1/ri,  from  (8)  and  (9)  gives: 


=  Kr     (10) 

Equating  (5)  and  (6)  when  2a  =  180,  t  =  tM  and  S  =  SM,  and  solv- 
ing for  pN,  which  is  constant,  gives: 

2  /-,i\ 

Value  of  t  at  any  Section.  —  Substituting  (5)  and  (7)  in  (8)  when  2a  = 
180,  and  again  when  it  equals  any  angle,  gives: 

1        l_       24pNr*       24pNr*  sin2  a 
r       r,  ~     EtM*  Et* 

from  which: 

t  =  tM\/^^  (13) 

Table  87  is  given  to  facilitate  calculation. 


PISTONS 
TABLE  87 


521 


2a 

sin  a 

sin2a 

\/sin2a 

20 

0.17365 

0.0302 

0.3112 

40 

0.34202 

0.1172 

0.4892 

60 

0.50000 

0.2500 

0.6300 

80 

0.64279 

0.4130 

0.7450 

100 

0.76604 

0.5870 

0.8375 

120 

0.86603 

0.7500 

0.9085 

140 

0.93969 

0.8830 

0.9595 

160 

0.98481 

0.9700 

0.9900 

180 

1.00000 

1.0000 

1.0000 

It  is  usually  accurate  enough  to  draw  a  circle  touching  as  many  of  the 
points  found  by  (13)  as  possible. 

From  (8),  conveniently  taking  S  =  SM  and  t  =  tM,  the  radius  of  the 
ring  when  free  is: 

r,  =  _ =  - =  mr  (i4\ 

1  *?  ^f  9  ^f  / 

~r~~  WM  ~  ~KE 

Length  of  Cut. — Referring  to  Fig.  354,  the  length  cut  out  of  the  ring 
after  it  is  turned  to  radius  TI  is: 

2?r(ri  —  r) 

This  allows  it  to  fit  in  the  cylinder  when  finished,  after  which,  a  small 
amount  c  may  be  cut  out;  the  total  cut  is  then: 

I  =  2rr(ri  -  r)  +  c  =  27rr(m  -  l)  +  c  (15) 

The  amount  c  cut  out  after  the  ring  is  turned  or  ground  to  exactly  fit  the 
cylinder  may  be  taken  as: 

c  =  0.006D  +  0.02"  (16) 

which  is  an  empirical  formula. 

Maximum  Stress. — For  a  ring  of  given  dimensions  the  maximum 
stress  in  place  is,  from  (8) : 

KM  ft        ti  (17) 


And  when  being  placed  in  the  groove: 

I 


1 


(18) 

To  assist  in  calculation,  or  to  determine  different  relations  approximately, 
Table  88  has  been  calculated  for  eccentric  rings.  The  stresses  are  high, 
but  for  the  grade  of  cast  iron  used  for  rings,  are  not  excessive. 


522  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  88 


a+*>?      •' 

K 

If  E  =  12,000,000  and  AC  =  1  .2 

SM 

icSAf 

PN 

m 

0.000915 

0.0436 

5,000 

6,000 

0.79 

.0194 

0.001095 

0.0480 

6,000 

7,200 

1.15 

.0212 

0.001280 

0.0519 

7,000 

8,400 

1.57 

.0228 

0.001465 

0.0555 

8,000 

9,600 

2.05 

.0246 

0.001650 

0.0592 

9,000 

10,300 

2.63 

.0258 

0.00183 

0.0623 

10,000 

12,000 

3.23 

.0270 

0.00201 

0.0654 

11,000 

13,200 

3.92 

.0290 

0.00219 

0.0684 

12,000 

14,400 

4.68 

1.0300 

0.00238 

0.0714 

13,000 

15,600 

5.52 

1.0310 

0.00256 

0.0741 

14,000 

16,800 

6.41 

1.0325 

0.00274 

0.0767 

15,000 

18,000 

7.35 

1.0335 

0.00293 

0.0795 

16,000 

19,200 

8.43 

1.0347 

0.00311 

0.0819     (       17,000 

20,400 

9.51 

1.0361 

Rings  of  Uniform  Section,  although  cut,  sprung  together,  and  turned 
to  fit  the  cylinder,  do  not  exert  a  uniform  pressure.  To  do  so,  r,  p,  E 
and  t  must  be  constant,  and  (12)  gives: 

1  1 


Et* 


a 


-  —  q  snv 


1  - 


24 PAT  sin2  a 
K*E 


-    1  - 


sin2a 


(19) 


KE 


The  value  of  TI  varies  at  every  point,  but  Unwin  gives  an  approximate 
method  for  finding  the  form  of  the  free  ring  by  assuming  7*1  the  same  over 
an  angle  of  30  degrees.  Fig.  358  shows  the  method,  in  which  abf  =  aa' 
b'b"  =  a' a",  etc.  The  distance  ca,  =  r\  for  the  arc  ab',  c'b'  for  the  arc 
b'b",  c"b"  for  the  arc  b"b'" ',  etc.  The  method  is  exact  if  these  arcs  are 
indefinitely  small,  so  that  greater  theoretical  accuracy  would  be  obtained 
by  taking  a  larger  number  of  divisions;  but  the  errors  of  draftsmanship 
increase  with  too  small  divisions  however,  offsetting  the  theoretical 
advantage. 

Large  rings  are  sometimes  hammered  to  this  form,  but  more  com- 
monly it  is  neglected.  All  rings,  however,  should  be  turned  or  ground  to 
the  diameter  of  the  cylinder  when  sprung  together.  A  common  method 
of  making  small  rings,  whether  eccentric  or  of  uniform  thickness,  is  to 
cast  a  pipe  and  cut  the  rings  from  it.  A  more  refined  way,  and  one  which 
will  not  cut  away  the  best  metal,  which  is  always  nearest  the  skin,  is  to 


PISTONS 


523 


cast  the  rings  separately,  allowing  but  slight  finish,  then  grind  to  size. 
Number  of  Rings. — An  empirical  formula  for  width  or  ring  is: 

w  =  0.02D  +  0.2"  (20) 

which  may  be  used  as  a  guide.     As  the  width  increases  but  slowly  with  the 


cylinder  diameter,  it  is  customary  to  retain  the  same  width  for  several 
diameters,  advancing  by  sixteenths  or  even  eighths  of  an  inch  for  the 
larger  sizes.  The  chart  of  Fig.  359  is  plotted  from  (20)  with  the  changes 
mentioned. 


"4 
I" 

4 

•? 

Trim 

1 

i" 

4 
0 

I 

I 

10"                10"                30  ' 
0 

FIG.  359. 

4 

50 

If  p  is  the  maximum  unbalanced  pressure  per  square  inch  in  the 
cylinder,  the  following  relation  may  be  of  assistance  in  fixing  conditions: 


pN  wn  = 


40 


from  which: 


=  JL  (21) 

This  is  an  empirical  formula  in  which  n  is  the  number  of  rings.     There  is 


524 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


no  definite  rule  for  determining  the  number  of  rings.  If  pN  is  assumed, 
n  may  be  found;  if  n  is  assumed,  pN  may  be  found,  and  from  Table  88, 
K  and  m  may  be  taken  for  determining  the  thickness  and  free  diameter. 
It  is  obvious  that  great  refinement  in  these  selections  is  not  necessary. 

It  may  be  considered  that  n  is  2  for  steam  engines,  and  should  a  single 
ring  be  used  the  width  may  be  twice  as  great;  otherwise  the  thickness 
will  be  excessive.  As  already  stated,  single  rings  are  provided  with 
springs,  so  that  a  thinner  ring  may  be  used,  giving  greater  flexibility 
and  a  more  uniform  pressure  against  the  cylinder  wall. 

Internal-combustion  pistons  have  from  3  to  10  rings,  3  being  a  com- 
mon number  for  small  engines.  Many  Diesel  engines  have  6  rings. 

Rings  must  be  fitted  into  the  grooves  so  that  they  are  free  to  adjust 
themselves.  F.  A.  Halsey,  in  the  Handbook  for  Machine  Designers, 
says  that  a  little  side  play  in  piston  rings  is  desirable,  and  that  the  steam 
pressure  will  force  the  ring  against  the  side  of  the  groove,  preventing 
leakage. 

DESIGNS  FROM  PRACTICE 

175.  Steam  Engine  Pistons. — Fig.  360  shows  a  built-up  piston  used 
on  the  Bass-Corliss  engine.  A  modification  is  made  in  the  original  design 


FIG.  360. — Bass-Corliss  piston. 

by  making  the  follower  bolts  as  described  in  Par.  173.     While  Babbitt 
metal  is  shown  in  the  bull  ring,  this  is  not  standard. 

Using  the  method  of  stress  determination  described  in  connection 


PISTONS 


525 


with  Fig.  351,  the  stress  from  (1)  is  2400  Ib.  with  100  Ib.  steam  pressure. 
With  125  Ib.  the  stress  is  2980  Ib.  It  is  probable  that  the  actual  stress 
is  not  so  high ;  at  -any  rate  this  piston  has  been  successfully  used  with  a 
pressure  of  125  Ib.  But  it  is  more  satisfactory  to  have  (1)  give  a  lower 
stress,  and  for  this  reason  Formula  (3)  was  devised  by  the  author  and 
gives  a  greater  depth.  The  change  makes  but  little  difference,  however, 
in  this  particular  design,  still  giving  2875  Ib.  stress  for  125  Ib.  pressure. 
With  the  ultimate  strength  of  cast  iron  16,000  Ib.,  this  gives  a  factor  of 
safety  of  but  5.57,  less  than  that  suggested  for  repeated  stress.  The 


Cyl.  Bore  less  JJM  _ 
'  Nominal  5ore  20" 


FIG.  361. — Mclntosh  and  Seymour  piston. 

possible  factor  for  reversed  stress  in  cast  iron  suggested  in  connection 
with  Table  72,  Chap.  XXI,  is  7.2.  With  this  as  a  basis  the  factor  of 
judgment  is  0.77,  or  less  than  unity,  which,  in  view  of  the  fact  that  the 
follower  plate  and  bull  ring  were  ignored  in  the  calculation,  may  be 
reasonable.  If  the  piston  spider  were  a  steel  casting  the  factor  of  judg- 
ment is  1.74;  it  would  probably  equal  at  least  unity  for  semi-steel  or 
some  of  the  better  grades  of  cast  iron  often  used.  Unwin  gives  3000 
Ib.  as  the  working  stress  in  pistons. 

Fig.  361  shows  a  piston  designed  for  Type  F  gridiron- valve  engines 
built  by  the  ]\JcIntosh  and  Seymour  Corporation,  Auburn,  N.  Y.     It  is  a 


526 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


box  piston  containing  no  ribs,  but  having  a  wide  face  and  rather  thick 
walls.     The  holes  through  which  the  cores  were  held  are  drilled  out  and 


FIG.  362. — Locomotive  piston. 

tapped  with  a  2-in.  pipe  tap,  plugged,  and  then  these  plugs  are  faced 
flush  with  the  piston  as  it  is  finished  in  the  lathe. 


FIG.  363. — Locomotive  piston. 

For  strength  calculations,  this  piston  is  composed  of  two  flat  plates; 
one  of  these  plates  has  a  central  load,  and  this  is  resisted  ai  the  edge  by 


PISTONS 


527 


a  portion  of  the  pressure  on  the  other  plate  transmitted  through  the 
cylindrical  portion  of  the  piston.  The  other  plate,  exposed  to  a  uniform 
steam  pressure  has  a  central  load,  and  at  the  edge  a  load  caused  by  the 
resistance  of  the  first  plate  to  bending.  Morley's  Strength  of  Materials 
contains  the  elements  of  an  analysis  of  this  condition,  and  it  may  be 


— 

H2  1 

-4 

pi 

T 

1 

i 

I     • 

1 

I 

1 

1 

i 

i 

1 

I 

1 

1 

Ol 

.i 

(L 

II 

ij 

1  1 

i 

1  1 

1  1 

1  1 

1  1 

-M 

1| 

P 

©       <s>      • 

1 

i 

L,. ,__^..._ 9-- - >J 


V-.OOS'Smaller  than  Sore  of : Cylinder— - 


FIG.  364. — Bruce-Macbeth  gas  engine  piston. 

possible  to  combine  them;  it  is  probably  not  practical,  however,  and  the 
design  must  be  mostly  empirical. 

Figure  362  shows  a  locomotive  piston  designed  by  the  American  Loco- 
motive Co.  It  is  a  conical  piston  with  cast  steel  body  riveted  to  a  cast 
iron  bull  ring.  With  a  pressure  of  200  lb.,  a  working  stress  of  5000  Ib. 


528 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


5ech'onA-A 


FIG.  365. — Franklin  automobile  engine  piston. 


FIG.  366. — Water-cooled  gas  engine  piston. 


PISTONS  529 

and  using  the  mean  angle  (90°  -  20.5°),  T  equals  1%  in.  This  is  but 
J£  in.  more  than  the  dimension  given. 

Another  style  of  locomotive  piston  by  the  same  builders  is  shown  in 
Fig.  363.  This  is  a  box  piston  with  ribs,  the  outer  ends  of  which  are  cut 
away,  allowing  the  core  to  extend  around  the  piston.  The  core  holes  are 
filled  by  driving  in  rivets  which  are  countersunk,  and  faced  in  finishing 
the  casting.  This  piston  is  cast  iron.  From  (1),  the  stress  with  200  Ib. 
pressure  is  1340  Ib.  Using  the  standard  factor  7.2,  the  factor  of  judg- 
ment is  1.66.  If  the  standard  factor  12  is  used  (see  Chap.  XXI),  the 
factor  of  judgment  is  unity. 

176.  Internal-combustion  Engine  Pistons. — A  gas  engine  piston 
used  by  the  Bruce-Macbeth  Engine  Co.,  Cleveland,  Ohio,  is  shown 
in  Fig.  364.  There  is  little  opportunity  for  strength  calculation  in  such 
pistons,  and  the  determination  of  their  dimensions  is  largely  a  matter  of 
judgment  based  on  experience.  The  bearing  surface  must  be  sufficient  to 
keep  the  bearing  pressure  within  certain  limits  and  this  is  considered  in 
Chap.  XXVI. 

In  the  larger  engines  running  at  slow  and  medium  speeds,  the  inertia 
effect  is  not  great,  and  sufficient  thickness  may  be  used  so  that  ribs  are 
not  necessary. 

Figure  365  is  the  piston  of  the  automobile  engine  built  by  the  Franklin 
Automobile  Co.,  Syracuse,  N.  Y.  It  is  very  fight,  and  ribbed  for  strength 
and  stiffness.  A  helical  groove  is  turned  upon  the  surface  for  oil  dis- 
tribution. There  are  practically  no  strength  calculations  which  may  be 
made  for  this  piston. 

A  water-cooled  piston  for  a  double-acting 'gas  engine  is  shown  in  Fig. 
366. 


CHAPTER  XXIV 

PISTON  RODS 
Notation. 

D  =  diameter  of  cylinder  bore  in  inches. 
d  =  diameter  of  body  of  piston  rod  in  inches. 
r  =  radius  of  gyration  of  rod  section.  ~* 

ti  =  thickness  of  wall  of  crosshead  neck,   considered  as  a  thick 

cylinder. 

to  =  ditto  for  piston  hub. 

p  =  maximum  unbalanced   pressure  per  square  inch  in  cylinder. 
Pi  =  normal  pressure  per  square  inch  in  rod  fit,  due  to  taper  or 

forcing. 
P  =  total  maximum  unbalanced  pressure  on  piston.     Also  used  for 

total  ram  pressure  in  making  pressed  fit. 
PA  =  portion  of  total  piston  load  taken  by  taper  when  rod   is  in 

compression. 

PF  —  portion  taken  by  friction  due  to  fit. 
PB  =  portion  taken  by  shoulder. 
ST  =  tensile  stress  through  key  eye  at  diameter  dz. 
SR  =  tensile  stress  at  root  of  thread. 
Sc  =  compressive  stress  in  portion  of  key  in  rod,  and  in  rod,  due  to 

key. 

SK  =  compressive  stress  at  shoulder. 
$1  =  hoop  stress  in  neck  of  crosshead  due  to  taper  fit. 
S0  =  hoop  stress  in  hub  of  piston  due  to  taper,  or  forced  straight  fit. 
Z  =  length  of  piston  rod  in  inches,  taken  from  center  of  piston  face 

to  center  of  crosshead  pin. 
n  =  taper  from  axis  in  inches  per  foot. 
p  =  coefficient  of  friction. 
/  =  factor  of  safety. 
m  =  number  of  threads  per  inch. 
T  =  tons  per  inch  of  diameter,  per  inch  of  length,  required  to  make 

a  pressed  fit — the  maximum,  as  fit  is  complete. 
TI  =  tons  per  inch  of  diameter  required  to  make  a  pressed  fit. 

530 


PTSTON  RODS  531 

177.  Piston  Rod  Formulas. — There  are  various  formulas  for  deter- 
mining the  diameter  of  piston  rods,  most  of  which  take  no  account  of  the 
method  of  attachment,  which  is  usually  the  weakest  part  of  the  rod.  Some 
formulas  consider  the  length,  and  while  it  is  well  to  check  the  rod  as  a  strut, 
in  few  instances  will  it  be  found  weak  in  this  respect  when  end  fastenings 
have  been  properly  provided  for.  Too  elaborate  an  analysis  may  not  be 
necessary,  but  from  an  examination  of  a  number  of  piston  rod  failures, 
it  is  evident  that  certain  important  points  are  sometimes  overlooked. 

There  are  many  methods  of  fastening  the  rod  to  crosshead  and  piston. 
The  best  crosshead  connection  in  the  author's  opinion  is  the  taper  fit 
and  key  when  properly  designed  and  constructed,  with  not  too  steep  a 
taper,  but  one  which  will  usually  allow  the  removal  of  the  rod  without  a 
hydraulic  press.  This  also  applies  to  the  piston  end  when  piston  design 
permits  of  its  convenient  application,  but  as  this  is  seldom  so,  the  taper 
fit  and  nut  may  be  used.  The  nut  may  be  circular,  partially  or  wholly 
concealed  within  the  piston,  and  tightened  by  a  special  wrench,  but  the 
plain  nut  is  simplest,  and  while  it  may  increase  the  clearance  volume 
slightly,  the  effect  upon  economy  is  negligible.  Standard  thread  is  too 
coarse  for  large  diameters,  so  to  facilitate  tightening,  a  finer  thread  is 
usually  employed;  five  or  six  threads  per  inch  gives  good  results. 

Steep  taper  with  no  shoulder  is  sometimes  employed,  but  the  shoulder 
is  preferable,  as  it  definitely  fixes  the  length  of  the  rod.  In  some  designs, 
as  in  Fig.  371,  the  end  of  the  rod  " bottoms"  in  the  crosshead,  an  excel- 
lent design  but  more  difficult  to  fit. 

A  straight  fit  with  shoulder  and  nut  is  sometimes  used  for  the  piston 
end,  and  is  good  practice;  but  a  pressed  fit  with  rod  end  cold-riveted  is  to 
be  condemned  as  unreliable. 

As  the  formulas  derived  will  cover  most  forms  of  attachment,  the 
analysis  first  employed  will  be  for  a  rod  having  a  taper  fit  and  key  at  the 
crosshead  end,  and  a  taper  fit  and  nut  at  the  piston  end.  It  is  assumed 
that  when  the  rod  is  in  tension  the  entire  load  is  carried  by  the  smallest 
rod  section  cut  by  the  key  at  the  crosshead  end,  and  by  the  section  at  the 
root  of  the  thread  at  the  piston  end.  In  compression,  at  both  ends,  the 
load  is  in  part  taken  by  the  wedging  action  of  the  taper  which  is  resisted 
by  the  hub  considered  as  thick  cylinder;  by  the  friction  of  the  wedging 
action;  and  the  remainder  by  the  shoulder. 

The  crosshead  end  is  shown  by  Fig.  367,  all  dimensions  being  in  inches. 
The  area  wd2,  subjected  to  crushing,  is  not  more  that  one-half  of  the 
area  2wb,  in  shear;  therefore,  crushing  will  be  considered  and  shear  ig- 
nored in  the  present  discussion.  With  low  compressive  stress,  shearing 
is  not  likely  to  begin. 


532 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


For  equal  strength  in  tension  and  compression,  for  the  most  econom 
ical  dimensions: 


or: 


Also: 


from  which: 


w  = 


STS 


p(S 


FIG.  367. 

Substituting  in  the  expression  for  w  gives: 


From  Fig.  367: 


irg 

4 

=  d, 


Ifb 


r6  v 

_di 
1  - 


nx 


Substituting  the  value  of  dz  just  found,  gives: 

J    _        D         lp(Sr  +  Sc) 
nx\       STSc 


-f 


(2) 


If  a  thrust  PA  will  produce  an  allowable  normal  unit  pressure  pi: 

PA  =       -T-dilipi. 


PISTON  RODS  533 

The  additional  thrust  which  may  be  resisted  by  friction  under  pressure  pi 
is: 

PF  =  Tr 


Then:  PA  +  PF  =  irdilipi  ( y^  +  /*)• 

If  this  is  equal  to  P,  no  shoulder  is  required,  but  if  less,  the  remainder, 
PB,  must  be  carried  by  the  shoulder.     Letting  n/12  -f-  M  =  k: 

PB  =  p  -  (pA  +  pF)  =  ^(D*p  -  4diZifcpi). 
Also: 

Equating  these  two  values  of  PB  and  solving  for  d  gives: 


It  will  be  found  convenient  to  determine  PI  for  a  given  value  of  /Si, 
the  allowable  stress  in  the  crosshead  neck.  From  (16)  and  (17),  Chap. 
XXI,  the  thick  cylinder  formula  for  free  ends  is: 

+  0.7pi  , 


-  13pi 

or: 


Pi  =-       2/  -  •  Si  =  gSi  (5) 

1.3(-^  +  1)  +  0.7 


Substituting  (5)  in  (3)  gives: 


d  .  p2?>  -  y.t.*ga.  +  dl.  (6) 

\  ^>X 

If  <i  is  required  for  known  rod  dimensions,  solving  for  q  from  (6)  gives  : 


And  from  (5)  : 


Solving  for  <t  in  (8)  gives: 


the  value  of  ^  being  found  from  (7). 


534  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

If  all  dimensions  are  known,  (7)  gives: 

Si  =  D2p  ~As]K1j  ~  dl^  (10) 

4rfi/i/Cf7 

Formula  (10)  may  be  used  for  investigation.  If  the  rod  has  no 
shoulder  and  is  not  bottomed,  SK  =  0;  then  (3)  and  (6)  fail,  and  q  or  Si 
may  be  obtained  from  (7)  or  (10),  and  ti  determined  from  (9).  A  steeper 
taper  is  then  required  to  prevent  excessive  hoop  stress. 

The  piston  end  of  the  rod  is  shown  in  Fig.  368.  The  same  analysis 
may  be  used  as  for  the  crosshead  end  when  the  rod  is  in  compression, 
replacing  d}  di,  Zi,  t  and  Si  by  dc,  d0,  10,  t0  and  S0.  Then  (6)  becomes: 


FIG.  368. 

Referring  to  (6),  (11)  and  (5),  it  may  be  seen  that,  if: 

d0  =  di 


and: 


and  if  the  taper  is  the  same  at  both  ends  of  the  rod,  dc  will  equal  d,  and 
the  collar  may  be  omitted  if  10S0  =  liSi',  or  if : 

8°  ^  h  (12) 

~Sl  ?  10 

As  the  piston  hub  is  strengthened  by  the  walls  and  ribs,  this  condition 
may  easily  be  assumed,  if  in  doing  so  the  resulting  value  of  dT  is  not  too 
small. 

To  find  dr,  the  diameter  of  the  screwed  end : 


PISTON  RODS  535 

Then: 

d,  =  D^  (12a) 

and: 

,         ,        1.3       n   [p     ,    1.3  , 

rfr     =    d3    +  :=     D^l—     +    _ 

'  r  t*  \  O  /£  *  *  t» 

Then: 

do  ?  dr  +  ^  +  0.125  (14) 

If  d0  is  greater  than  d\f  a  collar  may  be  forged  on  the  rod  as  shown  in 
Fig.  368,  of  diameter  dc,  which  may  be  found  by  (11) ;  or  safely,  and  more 
simply: 

dc  =  d0  +  d  -  di  (15) 

With  a  straight  forced  fit  in  the  piston,  n  equals  zero  and  k  =  /*, 
which  may  be  used  in  (11).  It  is  safe,  however,  to  take  k  =  0  in  (11). 
This  will  be  further  discussed  in  Par.  178.  If  a  key  is  used,  the  fit  diameter 
may  be  found  from  (2),  making  n  zero. 

Numerical  Values. — The  piston  rod  is  commonly  made  of  forged, 
open  hearth  steel,  known  as  machinery  steel,  the  elastic  limit  of  which  is 
given  as  38,000  in  Table  73,  Chap.  XXI.  In  the  key  eye  and  in  the  thread 
at  the  piston  end,  the  initial  stress  is  much  greater  than  that  due  to  the 
working  load  in  good  design  and  construction,  so  the  load  may  be  con- 
sidered static  with  a  standard  factor  of  safety  in  tension  of  2.  Due  to 
unknown  stresses  from  the  driving  of  the  key,  the  factor  of  judgment 
may  be  taken  from  2  to  2.5.  The  standard  factor  in  compression  is  3, 
but  as  the  stress  is  apt  to  be  more  evenly  distributed,  the  factor  of  judg- 
ment for  the  key  may  be  from  1.25  to  1.5. 

It  is  safer  to  assume  cast  iron  as  the  material  for  the  crosshead,  and 
as  the  load  is  practically  static,  the  standard  factor  is  4.  As  the  driving 
load  is  reversed — although  of  small  stress  value — the  factor  of  judgment 
may  be  from  1.5  to  1.75. 

The  factor  of  judgment  for  the  piston  hub  may  safely  be  unity,  due  to 
the  reinforcing  of  walls  and  ribs.  These  stresses  are  all  practically 
static,  and  are  not  affected  by  changing  loads  due  to  steam  and  gas  pres- 
sure and  inertia. 

As  lubrication  is  more  effective  with  a  taper  fit  than  with  a  straight 
fit,  the  coefficient  of  friction  may  be  0.1.  A  good  taper  is  1  in.  per  ft., 
total. 

From  the  foregoing,  and  from  other  considerations  of  good  design, 
the  following  data  may  be  taken : 


536  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

ST  =  SB  =  8000,  Sc  =  SK  =  10,000,  Si  =  2500,  So  =  4000,  n  =  0.5, 
fM  =  0.1,  b  =  di,e  =  c  =  Q.Sdi,  h  =  2.6di,  h  =  di/2,  £0  =  d0/2.  From 
the  above:  A;  =  0.1417,  x  =  1.8  and  q  =  0.509. 

The  crushing  stress  Sc  is  sometimes  taken  more  than  twice  as  great, 
in  which  case  it  is  necessary  to  check  for  shear;  but  the  lower  value 
reduces  unit  bending  load  on  the  key  considered  as  a  beam,  lessening 
deflection,  and  the  large  bearing  surface  minimizes  the  chance  of  damage 
due  to  driving  the  key. 

Substituting  the  above  values  in  the  various  equations  gives: 

d  =  0.0189Z>\/p  (16) 

di  =  0.0177£>\/P  (17) 

w  =  0.00524Z)y^  (18) 

li  =  0.046Z)Vp  (19) 

The  minimum  value  of  e  may  be  O.Gdi,  but  l\  must  not  be  decreased 
without  checking  by  (6).     The  distance  e  +  6  must  not  be  greater  than 
the  value  used  to  determine  di  and  d2.     It  is  well  to  make  b  ^  di. 
From  (12)  : 

Zo_>Si_2500 

li  ~  So  "4000  " 

If  10  ^  0.625/1,  no  collar  will  be  required,  and  d0  may  equal  di  as  in  Fig. 
369,  if  this  value  is  not  less  than  that  given  by  (14).  Then  from  (14)  : 


dT  >  d0  -         -  0.125  =  d0  ~          +  0.125)  (20) 

Also  from  (13)  : 

dT  ?  O.OllD-v/P  +  0.216  (21) 

The  greater  value  of  dT  must  be  used.     It  is  usually  preferable  to  have: 

d0  ^  di 

The  piston  rod  as  a  strut  may  be  checked  by  Formula  (36),  Chap. 
XXI.  The  rod  is  not  fixed  at  the  ends  as  this  would  involve  the  binding 
of  crosshead  and  piston  between  guides  and  cylinder  walls,  but  the  large 
bearing  surfaces  tend  to  keep  the  ends  of  the  rod  in  line,  and  for  safety 
it  may  be  considered  as  a  pin-ended  strut;  it  probably  is  between  a  pin 
end  and  a  flat  end  in  strength.  Formula  (42),  Chap.  XXI  may  then  be 
used  as  the  special  formula,  taking  I  from  the  center  of  the  piston  face  to 
the  center  of  the  crosshead  pin.  Solving  for  d  gives  (where  d  =  4r)  : 


'       ~~ 


The  factor  of  safety  /is  made  up  of  the  standard  factor  for  reversed  load, 
and  a  factor  of  judgment  of  from  1.4  to  1.6,  due  to  the  possibility  of 


PISTON  RODS  537 

some  eccentricity  from  adjustment  of  the  bull  ring.     Taking  /  as  9  for 
double-acting  steam  engines,  (22)  becomes: 


(23) 


If  this  value  is  greater  than  that  given  by  (6)  or  (16),  it  must  be  used; 
but  usually  it  will  give  a  smaller  value  and  may  be  used  when  lightness 
is  especially  desirable,  or  for  short-stroke  engines,  in  which  case  the  rod 
ends  must  be  proportioned  as  already  explained,  resulting  in  a  collar  at 
each  end. 

The  factor  f^T  for  double-acting  steam  engines,  for  parts  loaded  as  the 
piston  rod  body,  is  also  given  as  6  in  Par.  166,  Chap.  XXI,  making  the 
factor  of  judgment  1.5  for  the  value  just  assumed. 

Example.  —  As  an  example  of  design,  a  rod  will  be  proportioned  for 
the  20  by  48  in.  Corliss  engine  of  Chap.  XII,  the  unbalanced  steam  pres- 
sure being  125  Ib.  per  sq.  in. 

From  (16),  d  =  4.225,  say  4^"- 

From  (17),  di  =  3.96,  say  4". 

From  (18),  w  =  1.162,  say  1%6". 

From  (19),  Zi  =  10.29,  say  10^". 

Also  e  =  3.175,  say  3",  and  b  =  4". 

In  rounding  up  fractions  it  is  safe  never  to  make  di  less  than  the  value 
given  by  the  formula,  while  e  +  b  must  never  be  greater.  If  it  is  desired 
to  save  space,  e  may  be  made  %  of  the  calculated  value,  especially  with  a 
steel  crosshead. 

From  Chap.  XXIII,  the  piston  face  is  7  in.  (=  10),  which  is  greater 
than  0.625Zi,  and  no  collar  will  be  required.  Then  d0  =  di.  From  (20), 
dT  =  3.29,  say  3%  in.  From  (13),  dT  =  2.716  in.,  so  that  3J4  in.  from 
(20)  is  safe. 

Following  tentatively  through  the  design  of  the  details  involved,  the 
length  of  the  rod  I  is  about  86  in.;  then  from  (23),  d  =  3.82  in.,  showing 
that  the  rod  is  safe  as  a  strut. 

The  above  calculations  are  for  open  hearth  hammered  steel  with  an 
elastic  limit  of  38,000  Ib.;  for  other  steels  the  general  formulas  must  be 
used,  and  working  stress  substituted  according  to  the  judgment  of  the 
designer. 

178.  Pressed  Fits.  —  Straight  fits  are  sometimes  used  on  one  or  both 
ends  of  the  rod.  These  vary  from  a  sliding  fit  to  a  forced  fit  capable  of 
withstanding  the  entire  load,  and  in  some  cases,  with  the  addition  of 
riveting  over  the  end,  the  load  is  so  carried.  The  riveting,  however,  of 
cold  steel  adds  but  little  to  the  safety,  and  in  general  the  method  is 


538  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

untrustworthy.  If  supplemented  by  nut  or  key,  straight  fits  of  various 
degrees  of  tightness  are  satisfactory  when  once  assembled,  but  are  difficult 
to  take  apart  if  forced. 

A  simple  method  of  computing  the  stresses  due  to  forced  fits,  and 
perhaps  as  reliable  as  any,  is  to  assume  a  coefficient  of  friction,  and 
determine  the  relation  between  ram  pressure,  thickness  of  hub  wall, 
normal  pressure,  and  stress  resulting  therefrom.  The  method  was  in 
part  covered  by  the  taper  fit  analysis  already  given,  and  the  notation 
already  applied  to  the  crosshead  end  will  be  used,  di  being  the  uniform 
diameter  of  the  fit.  P  will  also  be  used  for  total  ram  pressure  and  T 
for  tons  per  inch  of  diameter  per  inch  of  length  of  the  fit,  due  to  P.  Then  : 


P  =  20007Wi  =  irdilipm  (24) 

From  which: 

T  =        P        =  ^^  (2V 

2MOdili      2000 

From  (5)  and  (24)  : 

P          200077         0  ,0_N 

pi  =  -yy-  =  -        -  =  qSi  (26) 

TTttiti/i  7TJU 

in  which,  from  (8)  : 

*-         '"1         <  -    (27) 

+0.7 


From  (26)  : 

p1=2000T=        P 

q  Si       '  ' 

Then  from  (9)  : 

di 


These  equations  give  the  relations  mentioned  and  are  general;  they 
will  be  referred  to  in  other  chapters. 

Taking  P  as  the  piston  load  and  writing  di  =  KD,  (25)  may  be  written : 

T  -        Dp  (30) 

'  2550KI, 

Or,  in  tons  per  inch  of  fit  diameter: 

»  T,  =  TL  = P  (31) 

f\  p  e  f\  TS~  vf  *  / 

From  (31)  and  (26) : 


PISTON  RODS  539 

It  is  obvious  that  TI  is  not  the  measure  of  stress  in  the  hub,  but  it  is 
nearly  always  the  quantity  given  in  limiting  pressed  fits.  If  li  is  constant, 
TI  varies  directly  as  the  cylinder  diameter,  but  this  is  usually  not  the 
case,  Zi  usually  changing  with  the  cylinder  diameter,  although  sometimes 
not  at  the  same  rate. 

While  it  is  proper  to  use  a  pressed  fit  for  piston  rods,  it  is  safer  not 
to  count  on  their  holding  power,  but  to  provide  a  key  or  nut,  with  a  collar 
to  take  the  entire  load.  An  example  will  illustrate.  Assume  a  20  in. 
engine  with  125  Ib.  steam  pressure,  and  for  the  piston  end,  changing  sub- 
scripts as  in  the  previous  paragraph,  let  d0  =  4  in.  and  10  =  7  in. 
Assume  a  stress  of  4000  Ib.  in  the  piston  hub  and  a  coefficient  of  friction 
of  0.25.  Let  the  outside  diameter  of  the  piston  hub  be  8  in. 

From  (27),  q  is  0.509;  from  (26),  pl  =  2036.  From  (25),  T  =  0.8 
and  from  (24),  P  is  44,800  Ib.,  the  ram  pressure  required.  The  maximum 
piston  load  is  39,250  Ib.  The  ram  thrust  is  some  greater,  but  as  this 
value  is  allowed  some  fluctuation,  it  may  be  assumed  as  about  the  same 
in  this  particular  case.  If  the  fit  is  to  carry  the  entire  load  there  must  be 
a  factor  of  safety  and  the  increase  of  stress  will  be  proportional  thereto. 
While  the  piston  walls  and  ribs  add  to  the  strength  of  the  hub,  the  uneven 
distribution  of  metal  and  the  possibility  of  shrinkage  strains  may  result 
in  cracks  if  the  stress  is  assumed  much  greater  than  would  be  permissible 
in  the  plain  hub  if  there  were  no  ribs.  With  a  factor  of  safety  of  only  2, 
the  stress  in  the  hub  would  be  8000  Ib.,  which  is  excessive  for  cast  iron. 
To  retain  the  stress  at  4000  Ib.,  T  being  1.6,  and  from  (26),  q  being  1.018, 
the  thick  cylinder  formula  fails,  showing  that  a  solid  piston  would  not 
permit  as  low  a  stress. 

The  situation  is  not  much  relieved  by  assuming  as  high  a  coefficient  of 
friction  as  0.4,  which  is  unwarranted,  as  the  surface  is  always  lubricated. 
It  looks  as  if  the  margin  of  safety  must  be  provided  by  undue  stress  and 
cold  riveting,  when  no  other  fastening  is  used.  It  seems  more  rational 
to  allow  a  reasonably  snug  fit  and  not  to  depend  upon  it  for  axial  load; 
then  dc  (Fig.  368)  may  be  determined  from  (11),  in  which  k  is  zero  for  a 
straight  fit.  The  diameter  of  the  threaded  portion  may  be  determined 
from  (14),  taking  n  as  zero  and  checking  by  (12a)  and  (13). 

The  rod  is  sometimes  fastened  to  the  piston  with  a  straight  fit  and  nut, 
and  screwed  into  the  crosshead,  where  it  is  held  by  a  lock  nut  or  clamped 
by  splitting  the  crosshead  neck.  As  the  ends  are  usually  safe  for  this 
design  if  the  rod  is  designed  as  a  strut,  Formula  (23)  may  be  used,  and  if 
it  is  desired  that  the  diameter  shall  be  the  same  for  all  strokes,  the  maxi- 
mum stroke  may  be  assumed. 

As  an  example,  let  a  rod  be  designed  for  the  20  by  48  in.  engine  of  the 


540 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


preceding  paragraph.  Using  the  same  data  as  far  as  it  applies,  I  =  86 
and  d  =  3.82,  usay  3J£  in.  Calling  this  dc  in  (11),  letting  k  be  zero  and 
solving  for  d0)  the  diameter  of  fit,  gives:  d0  =  3.162,  say  3J£  in.  The 
threaded  end  may  be  3  in.,  which  is  greater  than  the  value  given  by  (13). 
As  all  the  load  is  carried  by  the  shoulder  and  nut,  a  driving  fit,  or  even 
a  snug  hand  fit  will  be  ample.  It  may  be  considered  advisable  to  make 
the  fit  larger  and  use  a  collar.  If  the  fit  is  the  same  diameter  as  the  rod, 
the  diameter  of  the  collar  may  be  found  from  (11),  or,  dc  =  4.47,  say 
4J<2  in.  The  thread  may  be  3%  in.,  and  may  be  the  same  at  both  ends 
of  the  rod. 


TIG.  369. 

A  small  amount  may  usually  be  removed  from  the  rods  just  designed, 
for  the  purpose  of  truing  up  when  worn  out  of  round  or  scored,  so  it 
may  be  advisable  to  add  from  J{g  to  }/±  in. — depending  upon  the  rod 
diameter — to  the  diameter  of  the  body  of  the  rod  for  this  purpose,  es- 
pecially if  the  calculated  diameter  is  not  greater  than  given  by  (23). 
In  any  case,  the  shoulders  must  not  be  reduced  by  turning,  to  values  less 
than  those  given  by  (6)  and  (11). 

179.  Designs  from  Practice. — Fig.  369  shows  the  crosshead  end 
covered  by  Formulas  (16)  to  (19),  while  Fig.  370  shows  the  screwed  end. 
These  have  been  used  by  the  author  on  Corliss  engines  for  a  number 
of  years  and  have  been  entirely  satisfactory.  In  determining  the 
diameter  of  the  screwed-end  rod,  Formula  (16)  was  used  for  the  sake 
of  uniformity;  this  usually  gives  a  larger  rod  than  is  necessary  for  the 


PISTON  RODS 


541 


strength  of  the  crosshead  threaded  end,  and  if  the  screwed  end  is  to  be 
standard  construction,  (22)  may  be  used,  and  if  it  is  desired  to  use  the 
same  diameter  of  rod  for  all  strokes,  it  may  be  computed  for  the  maximum 
stroke  as  previously  stated. 


FIG.  370. 


With  the  smaller  rod  it  may  be  necessary  in  some  cases  to  use  the 
collar  at  the  piston  end,  shown  in  Fig.  368,  in  order  that  ds  may  not  be 
too  small.  This  may  readily  be  checked  by  (11),  and  the  diameter  of 
the  threaded  end  by  (21). 


FIG.  371. — Locomotive  piston  rod. 

Fig.  371  shows  a  locomotive  rod  used  by  the  American  Locomotive 
Co.  A  stress  of  9500  Ib.  in  tension  is  allowed  at  the  least  area  through 
the  key  way.  Considered  as  a  static  stress,  this  gives  a  factor  of  judg- 
ment of  2  if  the  elastic  limit  is  38,000  Ib.  Instead  of  a  shoulder,  this  rod 


FIG.  372. — Mclntosh  and  Seymour  piston  rod. 


''bottoms"  in  the  crosshead.  A  tail  rod  is  provided  which  connects  to  a 
small  crosshead  whose  guide  is  a  projection  from  the  cylinder  head.  The 
purpose  of  the  tail  rod  is  to  relieve  the  pressure  of  the  piston  due  to  its 
weight,  and  prevent  wear  in  the  cylinder. 


542 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


A  rod  used  by  the  Mclntosh  and  Seymour  Corporation  on  their 
Type  F,  Gridiron-valve  engine  is  shown  in  Fig.  372.  The  rod  is  screwed 
into  the  crosshead  and  held  by  a  lock  nut.  The  crosshead  for  the  same 
engine  is  shown  in  Chap.  XXVI,  and  the  piston  in  Chap.  XXIII.  The  fit 
in  the  latter  is  a  straight  hand  fit,  a  clearance  of  0.002  in.  being  allowed. 
This  fit  only  extends  a  short  distance  from  shoulder  and  nut,  the  interme- 
diate portion  being  turned  slightly  smaller.  A  special  nut  is  used,  and 


\ 

I 

n 

1 

1     »   o|    H 

FIG.  373. — Mclntosh  and  Seymour  piston  rod  details. 

is  sunk  into  the  piston  and  tightened  by  a  special  wrench.  The  wrench, 
nut  and  locking  collar  are  shown  in  Fig.  373.  The  collar  is  split,  and 
expanded  into  the  counterbore  of  the  piston  by  a  taper  set  screw.  The 
rod  shown  is  for  a  20  in.  engine;  its  dimensions  may  be  compared  with 
those  of  the  examples  given,  the  length  being  practically  the  same.  Rod 
and  nut  are  steel  forgings,  the  wrench  a  wrought  iron  forging  and  the  lock- 
ing plate  cast  iron. 

A  tandem  compound  rod  is  very  similar  to  the  rod  shown  in  Fig.  371. 


PISTON  RODS  543 

It  may  be  designed  by  the  foregoing  methods,  the  main  rod  taking  the 
maximum  thrust  of  both  pistons,  and  the  tandem  portion  the  thrust  of 
the  piston  farthest  from  the  frame.  The  low-pressure  cylinder  is  com- 
monly next  the  frame,  but  not  always;  by  this  arrangement,  the  main 
rod  affects  the  effective  piston  area  less.  The  strut  length  may  be  taken 
as  before  for  the  main  rod,  and  from  center  to  center  of  piston  faces  for 
the  extension.  If  an  intermediate  crosshead  is  used,  measurements 
will  be  to  the  center  of  its  pin.  The  rod  may  also  be  screwed  into  the 
crosshead,  and  straight  piston  fits  may  be  used. 

With  some  large  tandem  steam  engines,  and  with  most  large  gas 
engines,  an  intermediate  crosshead  is  used.  For  gas  engines  the  rod  is 
made  hollow  for  water  cooling,  but  the  same  general  principles  of  design 
ajjply  as  already  given. 


CHAPTER  XXV 

CONNECTING  RODS 
Notation. 

D  =  diameter  of  engine  cylinder  in  inches. 

d  =  diameter  of  round  rod  or  depth  of  section  in  inches,  in  plane  of 

vibration. 

6  =  width  of  section  in  inches. 

I  =  length  of  rod  from  center  to  center  of  pins  in  inches. 
R  =  radius  of  crank  circle  in  feet. 
L  =  stroke  of  piston  in  inches. 
A  =  area  of  rod  section  in  square  inches. 
/  =  moment  of  inertia  of  rod  section  about  axis  perpendicular  to 

plane  of  vibration. 

r  =  radius  of  gyration  of  rod  section  in  inches.  This  is  the  radius 
of  gyration  used  in  the  strut  formula,  and  does  not  necessarily 
correspond  to  the  value  of  /  just  given  (see  Formulas  (6)  and 

(9)). 

c  =  distance  from  neutral  axis  to  extreme  fiber  (=  d/2). 
W  =  weight  of  rod  in  pounds. 
w  =  weight  per  cubic  inch  of  rod  material. 
Wi  =  weight  per  inch  of  length  of  rod. 
M  =  bending  moment  in  pound-inches. 

F  —  force  in  pounds,  due  to  angular  vibration,  tending  to  bend  rod. 
P  =  total  force  exerted  by  piston  in  pounds. 
p  =  unbalanced  pressure  per  square  inch  in  cylinder. 
ps  =  ultimate  strength  of  rod  as  a  strut. 
SE  =  elastic  limit  of  rod  material. 
S  =  stress  in  rod  due  to  bending. 
SR  =  stress  resulting  from  bending  and  direct  load. 
a  =  acceleration  at  x  normal  to  center  line  of  engine. 
V  —  tangential  velocity  of  crank  pin  in  feet  per  second. 
.AT  =  revolutions  per  minute. 
n  =  ratio   of  length  of   connecting  rod  to  radius  of  crank  circle 

(=  1/12R). 

q  =  a  constant  in  strut  formula  which  varies  with  end  conditions 
(see  Chap.  XXI,  Par.  164). 

544 


CONNECTING  RODS  545 

fs  =  factor  of  safety  for  angular  vibration. 
fp  =  factor  of  safety  for  strut  load. 
e  =  ratio  of  width  to  depth  of  rectangular  section  (  =  b/d). 

180.  Body  of  Rod. — Empirical  formulas  are  often  used  to  determine 
the  dimensions  of  the  body  of  the  connecting  rod.  These  formulas  are 
suitable  within  a  certain  range  of  conditions  under  which  they  have 
proven  satisfactory  in  practice,  or  have  been  checked  by  a  more  complete 
analysis.  While  too  elaborate  an  analysis  is  neither  necessary  or  desirable, 
a  safe  rational  formula  must  take  account  of  the  combined  effects  of  the 
strut  load  and  the  vibration,  or  whipping  action.  While  methods  of 
checking  for  stresses  so  produced  are  given  in  certain  texts,  the  direct 
solution  is  omitted.  By  using  Johnson's  strut  formula  discussed  in  Chap. 
XXI,  the  diameter  or  depth  of  the  rod  maybe  solved  for,  and  by  numerical 
substitutions  a  reasonably  convenient  working  formula  derived. 

In  the  derivation  it  is  assumed  that  the  maximum  sum  of  the  unit 
load  at  or  near  the  center  of  the  rod  due  to  the  piston  thrust,  and  the 
compressive  bending  stress  due  to  whipping,  must  not  be  greater  than  the 
safe  unit  load  allowed  by  the  strut  formula.  This  may  not  be  strictly 
scientific,  but  it  seems  reasonable,  and  checks  well  with  more  exact 
analyses  of  combined  compression  and  bending.  It  is  further  assumed 
that  the  section  is  uniform  throughout  the  length  of  the  rod.  This  is 
no  doubt  always  on  the  safe  side  and  it  greatly  simplifies  matters. 


v*- 


FIG.  374. 

Fig.  374  is  a  sketch  of  a  simple  slider-crank  mechanism,  showing  the 
outline  of  the  rod  and  certain  notation  used  in  the  discussion. 

From  Formula  (31),  Chap.  XVI,  the  acceleration  at  right  angles  to 
center  line  of  engine,  of  any  point,  is  given  by: 

V2    x 
a --g- jane  (1) 

The  forces  resulting  from  this  acceleration  will  be  assumed  to  produce  a 
bending  moment  in  the  rod,  but  will  be  applied  to  the  actual  rod  length 
and  not  to  its  projection  upon  the  center  line.  No  error  of  practical 

35 


546 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


significance  is  involved  by  this  assumption.     Then  the  force  acting  per 
inch  of  length  is: 

MI  V2  x  .  . 
—  '  -D-  '  T  '  sin  0. 
g  R  I 

As  Wij  the  weight  per  unit  length  is  constant,  the  total  force  for  a  given 
value  of  B  is : 


sin  0  = 


WV2 
2gR 


sin  6 


(2) 


-,--J 

1, — J 


From  (1)  it  is  obvious  that 
the  load  F  due  to  inertia  is  distri- 
buted along  the  rod  directly 
proportional  to  the  distance 
from  the  center  of  the  crosshead 
pin,  as  shown  by  the  load  dia- 
gram in  Fig.  375.  The  distance 
of  the  center  of  gravity  from  the 
center  of  the  crosshead  pin  is 
2/31',  this  is  the  center  of  oscil- 
lation. The  reaction  on  the  crosshead  pin  is  F/3,  and  on  the  crank 
pin  2F/3.  More  exact  values  may  be  obtained  by  the  method  of 
Chap.  XVI. 

The  general  formula  for  bending  moment  is : 

M  =  moment  of  reaction — S  moment  of  loads. 

Then  for  the  crosshead  pin  reaction,  taking  moments  about  any  section 
distance  x  from  the  crosshead  pin : 


FIG.  375. 


F        _  wi    x 
3  '  x  "  <T  3 


sin  6  I  a-dx=  ^-— 7 
(ygril 

t/O 


wv 


(I2x 


sin  6 


(3) 


When  AT  is  a  maximum,  I2x  —  x*  is  a  .maximum.     This  occurs  when: 


Or  x 


dx   ' 

0.5771     Then  (3)  becomes: 
I       WV2 


M 


gR 


sin  6 


(4) 


Maximum  stress  due  to  combined  strut  and  beam  effect  will  occur 
between  0.5771  and  0.51.     When  the  rod  is  made  largest  at  the  center, 


CONNECTING  RODS  547 

and  tapers  toward  both  ends,  the  center  of  oscillation  and  section  of 
maximum  bending  moment  are  both  moved  nearer  the  center,  so  that 
the  section  of  maximum  combined  stress  is  but  a  small  distance  from  the 
center  of  the  rod.  If  the  rod  tapers  from  the  crosshead  end  to  the  crank- 
pin  end,  these  points  are  moved  toward  the  crank,  but  the  rod  section  is 
increased  in  that  direction.  In  either  case,  if  the  diameter  or  depth  of 
rod  section  determined  be  considered  as  at  the  center  of  the  rod,  it  is  safe 
to  assume  that  other  sections  will  be  ample,  especially  as  the  actual  weight 
will  be  less  than  for  a  rod  of  uniform  section  in  the  first  case,  and  the  in- 
creasing depth  of  section  provides  for  any  discrepancy  in  the  second. 
From  (11),  Chap.  XVI,  the  maximum  thrust  along  the  connecting 
rod  is: 

P 


mP. 


To  simplify  the  use  of  the  formula,  m  may  be  taken  as  l/\/l  —  l/n2> 
its  maximum  value;  or  a  somewhat  larger  value  may  be  used  to  cover 
all  cases. 

The  unit  load  on  the  rod  as  a  strut  is: 

mP  _  irmpD2 

The  unit  stress  due  to  bending  is: 

s  =  *f-c.    .     •      •    ;' 

Johnson's  strut  formula  for  ultimate  unit  load  from  (31),  Chap.  XXI 
is: 

Ps   —  ^E  ~ 

The  whipping  action  always  produces  reversed  stress  with  a  factor  of 
safety  fs;  while  for  direct  load,  the  stress  is  partly  reversed  with  single- 
and  double-acting  engines.  With  internal-combustion  engines  the 
pressure  factor  affects  the  factor  of  safety  and  this  is  explained  in  Par. 
166,  Chap.  XXI,  which  it  is  well  to  consult.  It  is  then  well  that  a  separate 
factor  of  safety  fP  be  used  for  piston  thrust,  making  it  more  convenient 
to  use  ultimate  loads  and  stresses  in  the  general  equation,  which,  ac- 
cording to  our  assumption  is: 

Ps  >  fsS  +  fptn  •  -j  (5) 


From  Table  82,  Chap.  XXI,  it  may  be  assumed  that  for  ordinary  cases 
the  factor  of  safety  for  double-acting  engines,  both  steam  and  gas,  may 


548  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

be  6,  and  for  single-acting  engines  3.  This  factor  3,  however,  is  the 
product  of  the  standard  f  actor  fA  and  pressure  f  actor  fT  of  Chap.  XXI,  and 
it  is  based  upon  the  maximum  steam  or  gas  pressure  being  used  to  deter- 
mine the  load.  Should  determinations  be  made  for  rod  dimensions  at 
different  crank  angles  for  internal-combustion  engines,  using  actual 
pressures  (including  inertia  of  reciprocating  parts),  the  proper  standard 
factor  fA  (Chap.  XXI)  should  be  used  instead  of  /P,  and  this  will  depend 
largely  upon  the  inertia  as  may  be  seen  in  Table  16,  Chap.  XXI.  For 
ordinary  work/P  may  be  taken  as  3,  and  P  as  the  maximum  total  pressure, 
determining  the  dimensions  when  the  crank  is  90  degrees  from  line  of 
stroke,  as  with  the  steam  engine,  in  which  maximum  thrust  and  maximum 
whipping  action  occur  when  in  this  position  if  cut-off  is  as  late  as  one- 
half  stroke,  and  engines  are  usually  so  designed  for  over-load  purposes. 
Substituting  in  (5)  the  values  already  given,  we  have: 

I      WV2c  irmpD2 

'' 


For  any  :form  of  section,  Where  d  is  the  diameter  or  depth  of  section,  we 
may  write: 

r  =  ad,        A  =  5d*,        I  =  <7d4,       c  =  g- 

Also  : 

L  7r2#W2       LW2    7       ,0   „       nL  w         1A       wdnLd2 

''  ==  12nR  -      >F  :=  wlA  -     — 


520 

Substituting  in  (6)  and  solving  for  d  gives: 


TrmfPpD         nL 

^  ^^  "  (7) 

Where: 


=  17,440,0005^    SU 

The  moment  of  inertia  /  should  be  taken  with  the  axis  at  right  angles 
to  the  plane  of  vibration.  From  the  strut  formula,  it  is  obvious  that  the 
resistance  to  failure  is  least  when  : 

^  =  maximum  (9) 

If  r  is  about  an  axis  in  the  plane  of  vibration,  the  end  conditions  may  be 
for  a  flat-ended  strut  in  high-grade  work,  but  more  safely  for  pin  ends. 
If  r  is  about  an  axis  at  right  angles  to  the  plane  of  vibration,  a  raund- 
ended  strut  should  be  taken,  as  the  pins  are  in  rotation  and  of  no  assist- 


CONNECTING  RODS 


549 


ance  in  guiding  the  strut;  in  fact,  large  pins,  poorly  lubricated,  may 
produce  a  bending  stress. 

The  length  of  connecting  rods  ranges  from  4  to  6  times  the  crank 
length,  the  smaller  values  being  used  for  gasoline  engines,  and  the  larger 
for  horizontal  stationary  steam  engines. 

Numerical  Values. — Usual  practice  is  to  make  the  connecting  rod  of  an 
open  hearth  steel  forging;  then  SE  =  38,000  and  w  =  0.284.  Since  the 
whipping  action  may  be  estimated  with  reasonable  accuracy,  the  factor 
of  judgment  may  be  unity;  then  fs  =  6.  For  double-acting  engines, 
fp  =  6  as  already  stated,  and  for  single  acting  engines  fP  =  3  for  usual 
work.  To  provide  for  a  slight  eccentric  load  due  to  movement  of  pins 
in  bearings,  we  may  make  m  =  1.1,  which  is  larger  than  any  value  of 
I/cos  <f>  found  in  practice.  Then  by  substituting  values  of  a,  d  and  <r, 
special  formulas  may  be  written  for  different  sections. 

For  a  circular  section: 

1  7T  7T 


<T    = 


64 


As  rupture  would  occur  about  an  axis  at  right  angles  to  the  plane  of 
vibration,  q  =  1.4;     Then: 


24,300,000,000 


And: 


d  =  K  +  A  K*  + 


34,500 


(10) 

(11) 


Formulas  (10)  and  (11)  are  also  applicable  to  turned  rods 
with  sides  planed  to  the  width  of  the  stub  ends,  when  the 
maximum  diameter  is  either  at  the  center  or  at  the  crank    x. 
end  of  the  rod. 

For  a  rectangular  section  let  d  be  the  depth  in  the  plane 
of  vibration,  and  6  the  width,  and  let  b  =  ed.     Then: 


5  =  e 


and 


(T    = 


12 


FIG.  376. 


Assume  round  ends  for  flexure  about  axis  xx,  Fig.  376,  and  flat  ends  for 
axis  yy.  From  Formulas  (39)  and  (42),  Chap.  XXI,  equal  strength  about 
both  axes  is  obtained  when: 

1.4      0.63 

2  r    2 

But: 

r*2=l2      and      ri>2  =  l2 


550 

Then: 

or: 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

1.4  =  0.63 

d2  =      b2 


e  =   ,  =  0.67. 
a 


Then  if  e  <  0.67,  the  rod  would  fail  about  axis  yy,  and  as  this  would 
usually  be  true: 

a  =  —4=     and     q  =  0.63. 
V12 

As  before,  substituting  in  (7)  and  (8),  special  formulas  for  rectangular 
sections  may  be  derived.     Then  : 


K  = 


32,400,000,000 


And: 


43,900e 


(13) 


For  an  I  section,  definite  proportions  must  be  assumed  in  order  to  de- 
rive a  special  formula.  By  trial,  a  section  may  be  selected  which  will 
give  a  theoretical  maximum  of  strength  per  Ib.  weight,  but  too  light  a  sec- 
tion may  have  a  tendency  to  buckle  or  twist,  even  though  the  usual  beam 
and  strut  formulas  may  indicate  sufficient  strength.  Thin  flanges  reduce 
the  value  of  r  about  the  y  axis,  which  must  be  the  axis  used  for  this 
-..-^^.^  section  in  selecting  q  and  r.  A  rather  rugged  section 

£  is  given  in  Fig.  377,  for  wh^ch  a  special  formula  will  be 

derived.     For  this  section : 


d  =  0.375,     o-  =  0.039,      a  =0.125,      q  =  0.63. 
Then: 


And: 


Fig.  378  gives  a  number  of  connecting  rod  /  sections  used  on  locomo- 
tives. 

Applications  of  Formulas  (11),  (13)  and  (15)  will  be  made  in  the  follow- 
ing examples: 

(1)  Find  the  diameter  at  center  of  a  round  rod  for  a  20  in.  by  48  in. 
Corliss  engine,  running  100  r.p.m.,  the  rod  being  6  cranks  long.  The 


CONNECTING  RODS 


551 


maximum  unbalanced  steam  pressure  is  125  Ib.  per  sq.  in.     From  (10): 

62  X  1002  X  483 


K  = 


From  (11): 
d  =  1.63  +A/1.< 


24,300,000,000 


1.63- 


,  (6  X  125  X  400)  +  (5.1  X  36  X  48*)       a  _  . 

=  6.5  m. 


34,500 

Trooien's  Formula,  Bulletin  of  University  of  Wisconsin,  No.  252, 
taking  no  account  of  speed  gives  5  in.  For  75  r.p.m.,  a  more  usual  speed 
for  the  older  Corliss  engines  of  the  same  size,  (11)  gives  5%  in.  diameter. 
The  above  shows  that  formulas  based  upon  former  practice,  taking  no 
account  of  speed,  are  not  generally  applicable.  It  is  also  apparent  that 
for  higher  speeds  a  rod  of  round  section  is  a  little  bulky. 


(<--- 


ST 

T  I 

« 

N. 

I        I 


Sack  End  Front  End 

Main  Roof 

FIG.  378. 


(2)  Find  a  rectangular  section  for  the  same  data,  taking  e  =  0.45. 
From  (12) : 


From  (13): 


d  =  1.23  + 


62  X  1002  X  483 
32,400,000,000 


(6  X  125  X  400)  + 


2.18  X  36  X  482 
0.45 


7.33  in. 


43,900  X  0.45 

This  gives  a  section  73  percent  of  the  area  of  the  round  one,  but  still 
rather  heavy. 

(3)  Find  the  depth  of  an  I  section  for  the  same  data,  of  the  form  of 
Fig.  377.    From  (14) : 

2  X  1002  X  483 


K 


From  (15): 
d  =  0.985 


40,400,000,000 


=  0.985- 


.  QS,2   ,   (6  X  125  X  400)  +  (4.37  X  36  X  48*)  _     . 
'•985  16,500 


552  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

This  gives  a  section  about  56  per  cent,  of  the  area  of  the  round  section  and 
77  per  cent,  of  the  rectangular  section. 

(4)  Find  the  section  at  center  of  a  gasoline  engine  rod.  The  engine  is 
3M  by  4  in.,  running  1500  r.p.m.  and  the  value  of  n  is  4.  The  section  is 
similar  to  Fig.  377,  but  with  the  following  values: 

a  =  0.224,         5  =  0.379,         a  =  0.0295,         q  =  1.4. 

The  combined  diagram  of  Fig.  187,  Chap.  XVI  is  used  to  determine  p,  and 
d  will  be  calculated  for  several  positions  from  (14)  and  (15).  As  actual 
pressures  are  used,  the  pressure  factor  of  Par.  166,  Chap.  XXI  is  not  re- 
quired. According  to  the  formulas  of  this  paragraph,  the  standard  factor 
for  this  case  is  3.5,  and  this  is  taken  as/P  in  the  formulas.  The  ratio  g/r2 
of  the  strut  formula  is  greater  about  the  x  axis,  so  is  used. 

Maximum  pressure  occurs  at  position  1  of  the  crank,  which  is  15 
degrees  from  the  head-end  dead  center.  The  depth  of  section  for  this 
position  is  0.828  in.;  at  position  2,  d  is  0.739;  at  3,  it  is  0.669.  It  is 
therefore  a  maximum  in  this  case  where  the  pressure  is  a  maximum,  the 
whipping  action  having  less  effect  than  the  decrease  in  pressure  as  the 
piston  moves  away  from  the  end  of  the  stroke. 

The  actual  depth  of  a  rod  for  an  engine  of  the  same  size  and  speed  is 
0.794  in.,  the  calculated  value  being  a  trifle  over  4  per  cent,  greater. 

Taking  the  approximate  method  previously  mentioned,  with  the 
method  of  Par.  166,  Chap.  XXI,  and  using  the  factor  of  safety  given  in 
Table  82  of  the  same  chapter;  then  assuming  the  maximum  whipping 
action  with  the  crank  90  degrees  from  line  of  stroke,  (14)  and  (15)  give 
d  =  0.883  in.  This  is  on  the  side  of  safety  and  a  little  over  11  per  cent. 
greater  than  the  depth  of  the  actual  rod.  In  this  method  it  was  assumed 
that  p  =  335,  the  maximum  gage  pressure,  and  //>  =  3. 

While  being  fairly  rational  and  providing  for  the  principle  straining 
actions,  Formula  (7)  gives  results  agreeing  with  practice. 

181.  Connecting  Rod  Ends.  —  Where  the  body  of  the  rod  joins  the 
"stub  end"  there  is  direct  repeated  or  reversed  stress  as  at  the  center  of 
the  rod.  There  is  also  bending  stress  due  to  vibration  and  this  may  be 
more  than  is  commonly  supposed,  although  often  neglected. 

Formula  (2)  may  be  written: 


Were  the  rod  of  uniform  section,  one-third  of  this  would  react  approxi- 
mately normal  to  the  rod  at  the  crosshead  end,  and  two-thirds  at  the 
crank  end.  The  reaction,  especially  'at  the  crank  end,  may  be  greater 


CONNECTING  RODS 


553 


than  this,  although  in  part  due  to  a  heavy  stub  end  which  is  partially 
balanced  and  may  exert  but  little  bending  moment. 

It  is  presumably  safe  to  assume  a  uniform  section  unless  the  desire  for 
extreme  lightness  necessitates  greater  refinement,  in  which  case  the  prin- 
ciples of  Chap.  XVI  must  be  resorted  to.  The  weight  is  then: 

W  =  0.142rcLA  (17) 

where  A  is  the  area  of  section  at  the  center  of  the  rod.  Taking  Fig.  375 
as  a  load  diagram  and  using  the  notation  of  Fig.  379,  the  bending  moment 
at  dimension  di  for  the  crosshead  end  is  given  by: 


At  the  crank  end  it  is: 


2Fhr.        11(        h.-t 

= -<r\_l  -  21®  -  T}  \ 


(18) 


(19) 


Formulas  (18)  and  (19)  assume  a  rod  length  from  center  to  center  of  pins, 
of  uniform  section.  While  the  moment  may  be  greater  than  given  by 
these  formulas  for  a  rod  with  the  greatest  depth  at  the  crank  end,  the 
moment  of  the  overhanging  end  of  the  stub — which  is  neglected — tends 
to  offset  this.  The  reaction  on  the  pin  is  not  necessarily  the  quantity 
with  which  to  compute  the  bending  moment  on  the  rod. 

At  dimension  dz  the  formulas  apply  by  substituting  12  for  li.  In  this 
case  the  resistance  is  of  two  beams  of  depth  d2.  Due  to  the  possibility 
of  more  than  one-half  of  the  moment  acting  on  one  strap,  the  factor  of 
safety  may  be  increased  somewhat.  The  stress  at  dz  due  to  the  steam  or 
gas  pressure  is  always  a  repeated  stress,  there  being  no  compression  be- 
side that  due  to  bending. 

As  assumed  in  deriving  the  formula  for  the  body  of  the  rod,  the  stress 
at  failure  may  be  taken  as  the  sum  of  the  product  of  each  stress  by  its 
factor  of  safety,  which  may  be  different  as  previously  explained.  Then : 

l.P=fM+fpmP  (2Q) 


554  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

where  AI  is  the  section  area  at  dimension  di.  This  formula  also  applies, 
at  dimension  dz,  and  at  both  ends  of  the  rod,  by  taking  the  proper  sub- 
script and  values  of  M.  For  the  strap  (at  d2)  kM  must  be  substituted  for 
M ,  and  kP  for  P,  where  the  minimum  value  of  k  is  0.5.  As  already  stated, 
it  is  better  to  assume  that  more  than  one-half  of  both  direct  and  bending 
load  is  applied  to  one  side  of  the  strap,  and  k  may  well  be  from  0.6  to  0.7. 
If  SR  is  within  the  ultimate  value  used  as  a  basis  for  applying  the 
factor  of  safety — which  is  the  elastic  limit  in  this  book  for  all  ductile 
materials — the  design  may  be  considered  safe. 

At   section  dz  the  form  for  a  turned  rod  is 
T     shown  on  Fig.  380.     The  modulus  of  section  about 
L     axis  xx  may  be  found  graphically  by  the  method 
given  in  Appendix  1,  or  the  depth  may  safely  be 
assumed  as  shown  by  the  dotted  line  in  the  figure. 
For  rods  of  circular  section  at  d\,  equation  (20)  may  be  written: 
di3  X  Xdi  X  Y  =  0 

Y  4fpwP 

in  which:  X  =  -    — ~— 

and         .  •      '  r.-ljg  I. 

Y2       X3 

the  solution  of  the  cubic  equation  gives: 


3I      Y  A     IY*  +  X* 

i  =  V  ~  2" +  VT  +  27" 


Y 

"  2  "    M  4     '27 

This  may  be  applied  to  both  ends  of  the  rod,  taking  proper  values  of  M, 
fs  and  /P,  and  taking  SR  =  S&. 

It  is  apparent  from  (21)  that  di  will  vary  with  the  distance  li  in  a 
very  complicated  manner.  If  there  were  no  direct  stress,  the  form  of  the 
rod  might  resemble,  to  some  extent,  a  cubic  parabola,  which,  if  li  were 
zero,  the  value  of  di  measured  at  pin  center  would  be  zero.  If  there  were 
no  bending  and  the  direct  stress  were  tension,  dt  would  be  constant 
throughout  the  length  of  the  rod.  The  theoretical  line  is  some  curve, 
and  the  required  diameter  becomes  smaller  more  rapidly  than  the  de- 
crease of  li.  It  is  therefore  safer  to  assume  a  value  of  l\  large  enough  so 
that  for  points  between  this  and  the  center  of  the  rod,  the  diameter 
calculated  from  (21)  will  lie  but  little  outside  the  conical  surface  between 
d\  and  d.  It  will  then  probably  be  safe  to  assume  that  approximately: 


CONNECTING  RODS  555 

For  rods  with  the  maximum  section  at  the  crank  end,  calculations  f  or 
di  are  usually  unnecessary  at  this  end.  With  the  value  of  l\  just  given, 
(18)  may  be  written  for  the  crosshead  end: 

Ml  =  0.033FI  =  O.QIQnFL  (22) 

Also  for  the  crank  end  (19)  becomes: 

Ml  =  Q.Q57FI  =  Q.02SnFL  (23) 

In  determining  d2,  the  moment  must  be  taken  about  the  point  giving 
the  maximum  value  of  $;  when  the  construction  is  as  in  Figs.  381  and 
382,  this  will  usually  be  at  the  center  of  the  wedge  bolt,  the  area  and 
modulus  of  section  both  being  reduced  here.  For  the  design  of  Fig.  383, 
the  maximum  moment  is  where  the  strap  joins  the  rod.  The  wedge  bolt 
does  not  weaken  the  section  at  this  point,  but  it  is  well  to  check  for 
direct  stress  alone  where  the  strap  is  cut  away  by  the  wedge  bolt. 

For  a  rectangular  section,  which  may  be  assumed  at  d2,  (20)  may  be 
written  : 


2      fPmkP  _ 

--  r-o  --  «2  --  r-«  —  =  0. 


From  which: 

,       fPmkP 


I  /fPmkP\  2     QfskM 
"  "  \  V  2b2SR  )  '  ~  b2SR 


where  k  has  the  value  mentioned  in  connection  with  (20). 

Formula  (24)  may  also  be  used  to  determine  rod  neck  dimension  di 
for  rectangular  sections  by  using  subscript  2  and  making  k  =  1.  It  is 
more  usual  with  rods  of  rectangular  section  to  have  straight  sides  tapered 
from  crosshead  to  crank  end.  Then  (24)  may  be  used  to  check  the  ends, 
but  (7)  or  (13)  must  be  used  for  the  depth  of  section  at  the  center. 

The  value  of  di  may  not  be  conveniently  solved  from  (20)  for  an  I 
section,  but  may  be  checked  by  it. 

The  wedge  bolts  were  assumed  rather  arbitrarily  in  the  formulas  of 
Figs.  381  to  383,  but  as  they  may  be  unduly  tightened,  they  were  made 
more  rugged  than  is  sometimes  done.  If  t  is  the  taper  in  12  in.,  the 
tension  on  the  bolt,  neglecting  friction  and  initial  tension  is  : 

PB  =  g  (25) 

A  single  threaded  bolt  with  nut  is  sometimes  used,  but  if  the  brasses  do 
not  butt  together  so  that  the  wedge  is  held  firmly  against  the  brass,  the 
two  tap  bolts  shown  in  Figs.  381  to  383  are  preferable,  as  they  lock  the 
wedge  more  firmly.  A  more  general  expression  for  wedge  bolt  tension 
is  given  by  (10),  Chap.  XXIX. 

The  formulas  of  this  paragraph  may  be  applied  to  the  rod  for  the  20 


556  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

by  48  in.  Corliss.  Figs.  381  to  383  may  be  considered  as  designs  of  the 
rod  ends.  From  (17),  as  d  =  6.5  in.: 

W  =  0.142  X  6  X  48  X  33.2  =  1360  Ib. 
From  (16)  and  (22)  the  moment  at  di  for  the  crosshead  end  is: 

M  =  21,300. 
From  (23),  at  the  crank  end: 

M  =  37,500. 

The  total  thrust  on  the  piston  P  is  39,250  Ib.  Then  taking  the  same 
values  of  }P,  fs,  m  and  S  used  for  the  center  of  the  rod,  X  =  —  8.69  at 
both  ends;  Y  =  —  34.4  at  the  crosshead  end  and  —60.3  at  the  crank 
end.  From  (21),  the  neck  diameter  at  the  crosshead  end  is  4. 12  in.,  and 
may  be  made  4J£  in.  At  the  crank  end  the  diameter  is  4.85  in.  and 
may  be  made  4%  in. 

For  the  dimension  d2  taken  at  the  wedge  bolt  at  the  crosshead  end, 
(20)  gives  SR  =  28,150  Ib.  for  the  dimensions  given  under  Fig.  381,  as- 
suming the  stress  equally  divided  between  the  upper  and  lower  side  of 
the  strap.  At  the  crank  end,  with  wedge  between  pin  and  rod  as  in 
Fig.  382,  SR  =  40,750  Ib.  If  the  wedge  is  at  the  outer  end,  SR  = 
24,650  Ib.  These  values  are  all  well  within  38,000  except  the  crank-end 
stub  with  wedge  between  pin  and  rod.  It  was  stated  that  in  (22),  k 
must  not  be  less  than  0.5,  and  may  be  made  0.6  to  0.7.  For  the  three 
values  of  stress  just  given,  k  is  0.675,  0.467  and  0.77  respectively.  From 
the  last  two  it  is  obvious  that  Fig.  383  is  a  better  design  than  Fig.  382  for 
a  round  rod  of  the  stroke  and  speed  assumed  in  the  example. 

The  wedge  bolts  are  1%  in.  in  diameter,  the  area  at  root  of  thread 
being  1.057  sq.  in.  The  wedge  taper  is  taken  as  1J^  in.  per  ft.  Then 
from  (25).  the  total  maximum  pull  on  the  bolt,  neglecting  initial  stress  is: 

39,250  X  1.5 


12 
And  the  stress  is: 


4900 Ib. 


which  is  low,  giving  a  factor  of  safety  of  8.4  with  an  elastic  limit  of 
38,000  Ib. 

182.  Designs  from  Practice. — A  number  of  designs  which  have  been 
successful  in  practice  will  be  illustrated.  By  assuming  a  standard  engine 
as  explained  in  Par.  63,  Chap.  XII,  and  Par.  72,  Chap.  XIII,  the  dimensions 
of  stubs  may  be  given  in  terms  of  pin  diameter.  This  has  been  done 
by  the  author  in  Figs.  381  to  383,  the  formulas  given  under  each  figure. 
These  designs  were  used  for  several  years.  The  notation  applies  only 


CONNECTING  RODS 


557 


to  the  figures.  The  strap  thickness  was  not  determined  by  (22),  but 
that  it  checks  well  with  it  may  be  seen  by  the  example  of  the  preceding 
paragraph. 


;X\H£B3 

V  ^SasC   ~f~* 

/      NL  .,11." 


FIG.  381. — Crosshead  stub. 

a  =  I.l5di         6  =  0.8J      c  =  0.9d!        e  =  0.15d  +  0.35       g  =  c  +  e 
=  0.3di          j  =  0.86^  fc  =  1.15d!    m  =  d,  +  0.8d  n  =  0.75dj 

t^  =  r  +  0.4di 


. gr J*. u J 

FIG.  382. — Crank  pin  stub. 

a  =  l.Ud          b  =  0.8Z      c  =  0.9^       e  =  0.15d  +  0.35         gr  =  c 
h  =  0.3^  j  =  0.86d    k  =  2.25d  m  =  l.Sd    n  -  0.75d!  5  =  1. 

r  =  g  +  l.le  t  =  ^  u  =  r  +  QAd 


FIG.  383. — Crank  pin  stub. 


558 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  formulas  for  Fig.  382  apply  in  general  to  Fig.  383,  any  changes 
being  obvious. 

The  stubs  are  for  solid-end  rods,  much  used  for  Corliss  engines,  and 


FIG.  384. — Mclntosh  and  Seymour  connecting  rod  details. 

others  of  about  the  same  class.  While  turned  rods  are  shown,  rec- 
tangular or  I  sections  may  be  employed,  in  which  case  m  is  the  depth  and 
not  the  diameter,  and  may  be  made  some  less.  The  stubs  were  designed 


1                         1 
10                     | 

1  1— 

I 

IO          o        1 
I          1  

FIG.  385. —  Mclntosh  and  Seymour  connecting  rod. 

for  a  maximum  unbalanced  steam  pressure  of  125  lb.,  and  a  pressure 
per  sq.  in.  of  projected  area  of  1200  lb.  for  the  crosshead  pin  and  1000  lb. 
for  the  crank  pin,  and  can  therefore  not  be  taken  as  generally  applicable. 


CONNECTING  RODS 


559 


While  Fig.  382  has  been  much  used,  Fig.  383  gives  a  stronger  stub, 
especially  for  high  speeds,  as  the  bending  arm  Z2,  Fig.  379,  is  less. 

The  boxes  are  made  of  brass  or  some  of  the  bronzes,  and  it  is  usual  to 
line  only  the  crank-pin  box  with  babbitt  metal.  The  box  flanges  are 
sometimes  omitted,  the  brass  being  flush  with  the  rod  end.  The  wedges 
and  adjusting  bolts  are  of  steel  and  it  will  be  noticed  that  they  are  of  the 
same  diameter  in  both  stubs,  e  being  in  terms  of  the  crank-pin  diameter 
in  both  cases. 

Oil  grooves  are  shown  in  Chap.  XI.  The  space  between  the  two  halves 
of  the  boxes  is  usually  left  open  in  stationary  engine  practice,  but  some 
engineers  believe  they  should  butt  together,  and  when  adjustment  is 


lr-1-j-nl 


FIG.  386. — Crosshead  stub  for  main  locomotive  rod. 

made,  they  may  be  filed  or  machined;  or  they  may  be  provided  with 
"shims."  In  either  case,  the  wedge  is  drawn  tight  against  the  box. 

It  is  claimed  that  the  design  shown  in  Fig.  383  is  preferable  to  that 
of  Fig.  382,  as  adjustment  is  in  the  same  direction,  tending  to  keep  the 
rod  length  from  center  to  center  constant.  Both  forms  are  standard. 

Figure  384  shows  details  of  the  crank-end  stub  used  on  the  Mclntosh 
and  Seymour  Type  F  steam  engine.  The  crosshead  end  is  of  the  same 
design  but  smaller.  The  pin  fits  are  the  same  length  and  both  crosshead 
and  crank  ends  are  babbitted.  It  will  be  noticed  that  the  wedge  half 
of  the  box  is  drilled  and  tapped  for  "spreading  bolts."  These  may  be 
adjusted  so  that  the  wedge  may  be  drawn  up  snugly  against  the  box, 
making  the  filing  of  the  brass  or  the  use  of  shims  unnecessary.  The  rod, 


560 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


which  is  rectangular  in  section  is  shown  in  Fig.  385.  The  arrangement 
of  wedges  is  that  of  the  combination  of  Figs.  381  and  383,  adjustment 
tending  to  keep  the  rod  of  constant  length.  There  is  a  single  through 
wedge  bolt  for  each  stub,  of  the  type  previously  referred  to. 


Figure  386  shows  the  crosshead-end  stub  of  the  main  rod  of  a  locomo- 
tive, built  by  the  American  Locomotive  Co.  The  brasses  butt  together 
which  is  usual  locomotive  practice.  There  are  no  flanges  on  the  brasses 
of  this  stub.  The  adjusting  wedge  draws  crosswise,  pressing  against  a 
steel  block,  which  in  turn  forces  the  brasses  together. 


CONNECTING  RODS 


561 


Figure  387  shows  the  connecting  rod  used  on  the  Franklin  automobile 
engine.  At  the  crosshead  end  the  rod  clamps  to  the  pin,  the  wearing 
surfaces  being  in  the  piston  hubs.  This  rod  is  a  drop  forging  of  channel 
section,  suitable  for  very  high  speeds. 

A  marine  rod  end  is  shown  in  Fig.  387.  A  design  of  this  type  adapted 
to  large  engines  is  shown  in  Fig.  388.  This  may  be  modified  in  detail. 


FIG.  388. — Marine-end  stub. 

When  used  for  double-acting  engines,  one  bolt  must  be  strong  enough  to 
carry  the  fraction  k  (see  Par.  181)  of  the  maximum  piston  load,  or  the  maxi- 
mum value  of  T,  Chap.  XXI,  Par.  166.  The  projection  from  the  rod  which 
holds  the  bolt  must  also  be  strong  enough,  considered  as  a  cantilever, 
to  take  the  same  load.  The  load  coming  upon  these  parts  may  be  de- 
termined by  the  equations  of  the  paragraph  just  referred  to. 

In  some  marine  steam  engines  having  a  four-bar  guide,  the  same  type 
of  stub  may  be  used  for  the  crosshead-end  of  the  rod,  but  usually  a  solid- 
end  rod,  similar  to  Fig.  381  or  Fig.  386  is  used  A  design  used  for  a  Diesel 
engine  is  shown  in  Fig.  389.  The  force  on  the  rod,  except  that  due  to 
inertia,  is  all  in  one  direction,  so  the 
adjustment  does  not  require  the  strength 
necessary  for  a  double-acting  engine. 
Most  Diesel  engine  rods  are  of  circular 
section  with  but  small  increase  in  diam- 
eter from  crosshead  to  crank  end.  The 
whipping  action  is  small  compared  to  the 
thrust  due  to  gas  pressure,  and  the  circular  section  is  a  good  form  for 
strut  loads. 

There  have  been  many  designs  of  rod  ends,  most  of  which  possess 
merit.  There  is  as  yet  no  standard,  but  perhaps  not  so  many  different 
designs  are  used  as  was  the  case  several  years  ago.  In  selecting  a  design, 
simplicity,  rugged  construction,  ease  of  adjustment,  and  small  liability  of 
parts  working  loose,  with  cheap  production,  will  be  the  criteria. 


FIG.  389. — Diesel  wrist-pin  stub. 


CHAPTER  XXVI 

CROSSHEADS 
Notation. 

p  =  maximum  unbalanced   pressure   per   square  inch  acting  on  the 

piston. 

P  =  maximum  nominal  pressure  on  crank  pin  per  square  inch  of  pro- 
jected area. 

PI  =  same  on  crosshead  pin. 
Ps  =  same  on  crosshead  shoe. 

PX  =  total  maximum  unbalanced  pressure  on  piston. 
PP  =  total  pressure  on  piston  at  any  part  of  its  stroke. 
PN  =  total  normal  pressure  on  guide  at  any  part  of  stroke  due  to  piston 

thrust  only. 
pA  =  total  normal  pressure  on  guide  due  to  piston  thrust  and  inertia 

forces,  as  given  in  Formula  (53),  Chap.  XVI. 
PR  =  total  resultant  pressure  on  crosshead  pin  due  to  piston  thrust  and 

inertia  of  moving  parts. 

S  =  bending  stress  in  crosshead  pin  in  pounds  per  square  inch. 
D  =  diameter  of  cylinder  in  inches. 
d  =  diameter  of  crosshead  pin  (in  bearing)  in  inches. 
I  =  length  of  crosshead  pin  bearing  in  inches. 

A  =  projected  area  of  crosshead  pin  bearing  in  square  inches  =  dl. 
L  =  length  of  crosshead  shoe  in  inches. 
w  =  width  of  crosshead  shoe. 
AS  =  area  of  crosshead  shoe  =  wL. 
M  =  bending  moment  on  crosshead  pin. 
n  =  ratio  of  length  of  connecting  rod  to  length  of  crank. 
/  =  factor  of  safety. 
/3  =  factor  of  judgment  (see  Chap.  XXI). 

183.  The  crosshead  is  virtually  a  knuckle  joint  between  the  piston 
rod  and  connecting  rod.  In  order  to  take  the  thrust  due  to  the  angularity 
of  the  connecting  rod,  shoes  are  provided  which  slide  on  the  guide. 
Usually  these  shoes  are  made  adjustable  to  provide  for  wear  between 
crosshead  and  guide  or  between  piston  and  cylinder;  however,  some  build- 
ers?  feeling  that  the  adjustment  is  liable  to  be  tampered  with  and  claiming 

562 


CROSSHEADS  563 

that  the  bearing  pressure  is  such  as  to  make  the  wear  negligible,  omit  the 
adjustment. 

It  does  not  seem  practicable  to  make  locomotive  crossheads  adjust- 
able; when  it  is  necessary  to  take  up  wear,  new  gibs  are  furnished  or 
shims  are  used. 

Material. — Cast  iron  has  been  much  used  for  crossheads,  but  many  are 
now  made  of  steel  castings.  The  shoes  may  be  of  cast  iron  even  when  the 
body  is  of  steel.  The  shoes  are  sometimes  faced  with  brass,  but  usually 
with  babbitt  metal.  The  pin  is  commonly  an  open  hearth  steel  forging 
and  fitted  to  the  crosshead  with  a  taper  fit.  For  other  steels  see  Par.  159, 
Chap.  XXI. 

Strength. — There  are  comparatively  few  strength  computations  for 
the  cross  head.  The  longitudinal  stress  is  a  reversed  stress  for  double- 
acting  engines  (see  Chap.  XXI,  Par.  166) .  The  factor  of  judgment  should 
provide  for  the  possibility  of  water  in  the  cylinder  for  steam  engines,  being 
from  1.25  to  1.5.  The  area  through  the  weakest  section  should  be  such 
that  the  stress  will  not  exceed  the  safe  working  stress.  For  cast  iron  this 
may  be  from  900  to  1200  Ib.  per  sq.  in.,  and  for  a  steel  casting,  from  3000 
to  4000  Ib.  Semi-steel  is  sometimes  used ;  this  is  generally  stronger  than 
cast  iron,  but  some  builders  allow  the  same  working  stress  as  for  good  cast 
iron. 

184.  Crosshead  Pin. — In  proportioning  the  wearing  portion  of  the 
crosshead  pin,  some  ratio  of  length  to  diameter  must  be  assumed.  If  in 
a  steam  engine  with  over-hung  crank,  the  length  of  the  pin  is  equal  to  the 
diameter,  it  is  convenient  to  make  the  length  of  the  crosshead  pin  the 
same  as  that  of  the  crank  pin,  and  its  diameter  inversely  proportional 
to  the  pressure  per  sq.  in.  of  projected  area  allowed  on  the  pins.  Then 
if  PI  is  the  pressure  per  sq.  in.  on  the  crosshead  pin,  and  P  the  pressure 
on  the  crank  pin;  if  I  and  d  are  length  and  diameter  respectively  of  the 
crosshead  pin,  the  projected  area  is: 

If  p  is  the  maximum  unbalanced  pressure  per  sq.  in.  in  the  cylinder 
and  D  is  the  cylinder  diameter  in  inches,  equating  the  maximum  piston 
thrust  with  the  total  pin  pressure  gives: 

=  PA  =  PI2  (2) 


=  DVT  p    °-887Np 

Then:  d  =  ^l  (4) 


564  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

These  equations  are  convenient  for  steam  engines  for  the  conditions 
assumed,  For  center-crank  engines  the  equations  ma}^  be  used  by 
assuming  tentatively  a  side-crank  engine  and  designing  the  crosshead  pin ; 
then  the  crank  pin  may  be  given  a  separate  treatment. 

Bearing  design  is  discussed  in  Par.  52,  Chap.  XI,  and  bearing  pressures 
for  different  cases  given  in  Table  19  of  the  same  chapter. 

If  the  crosshead  or  piston  were  assumed  perfectly  rigid,  the  crosshead 
pin  would  be  a  beam  fixed  at  both  ends  and  loaded  with  a  load  somewhere 

between  a  uniform  and  a  concentrated 
i  load.     It  is  probably  safer,  especially  with 

-Jj   .  trunk  pistons,  to  assume  the  pin  as  sup- 

y/////\        V///A        V/////A      ported  at  the  center  of  each  hub.     The 


load  may  safely  be  taken  as  concen- 
trated at  the  center  of  the  pin;  but  as 
this  is  extreme,  it  is  sometimes  necessary, 
in  order  to  keep  the  pin  diameter  from 
L  \ L  I  being  excessive,  to  make  different  as- 

sumptions— that  a  certain  fraction  of  the 

FIG.  390-  i          ,1        £    ,1  -  ,1  i        ! 

length  of  the  pin  carries  the  entire  load 

uniformly  distributed.  Calling  this  fraction  q}  and  Px  the  total  maximum 
pressure,  a  general  expression  may  be  found.  From  the  general  equation 
for  bending  moment,  and  referring  to  Fig.  390,  the  bending  moment  is: 


Equating  this  with  the  modulus  of  section  gives: 

0.098&Z3  =  -r-(lo  —  ql)  (5) 

As: 

'•;"  '•  * --^ '•••-.:•:•.  ^fo&££ 

Equation  (5)  may  be  written: 
or  if  S  is  assumed : 


S  =  °  (6) 


d  = 


The  length  10  may  be  taken  to  the  center  of  the  pin  hubs,  or  it  may  be 
varied  according  to  judgment.  For  crossheads,  the  author  has  often 
made  the  hub  length  one-half  the  pin  length  (in  the  bearing);  taking 
IQ  to  the  center  then  makes  IQ  =  1.51.  The  value  of  q  may  be  taken  as 


CROSSHEADS  565 

0.5.     If  the  pin  is  first  proportioned  for  bearing  surface  it  must  be  checked 
for  stress,  or  vice  versa. 

For  steam  engines  with  cut-off  as  great  as  one-half  stroke,  p  is  at  least 
equal  to  maximum  unbalanced  steam  pressure  whether  inertia  is  taken 
into  account  or  not,  as  near  mid-stroke  inertia  is  zero.  With  internal- 
combustion  engines,  the  maximum  gas  pressure  acts  only  near  dead  center. 
The  maximum  pressure  including  inertia  may  not  be  near  dead  center 
if  the  inertia  is  great,  but  this  should  be  found  from  a  diagram.  If 
inertia  is  considered  in  design,  it  should  not  include  the  connecting  rod 
for  calculations  on  the  crosshead  pin  and  it  is  well  to  check  for  pin  stress 
with  the  maximum  gas  pressure  alone,  as  this  condition  obtains  at 
starting. 

More  correctly  the  load  on  the  pin  is  the  resultant  of  Px,  FP  and  pA, 
the  last  two  being  given  by  Formulas  (53)  and  (32),  Chap.  XVI.  The 
effect  of  pA  is  small  (See  also  Fig.  204  and  Table  56,  Chap.  XVI). 

See  Par.  166,  Chap.  XXI  for  factor  of  safety.  A  factor  of  judgment 
of  from  1.25  to  1.75  should  be  employed  when  neglecting  inertia,  etc. 

185.  Crosshead  Shoe. — It  is  customary  to  limit  the  maximum  pres- 
sure per  sq.  in.  on  the  crosshead  shoe.  Values  of  this  limiting  pressure 
are  given  in  Table  19,  Chap.  XI.  The  pressure  on  the  guide  PN  for 
any  crank  angle  is  given  by  (12),  Chap.  XVI,  and  is: 

PP  sin  6 


/sn 
~  \  n 


where  6  is  the  angle  the  crank  makes  with  the  line  of  stroke,  measured 
from  the  head-end  dead  center;  n  is  the  ratio  of  connecting  rod  to  crank, 
and  PP  is  the  total  piston  pressure  at  any  point  in  the  stroke. 

For  steam  engines  with  a  cut-off  as  long  as  one-half  stroke,  PP  =  Px 
and  sin  9=1;  this  gives  the  maximum  guide  pressure.  '  Tbfen  (8) 
becomes: 

P  P* 

(9) 

If  n  =  4,  and  the  quantity  in  the  radical  is  ignored,  the  error  will  be 
about  3  per  cent.  With  larger  values  of  n  the  error  is  less. 

For  internal-combustion  engines  the  maximum  value  of  PN  is  not  so 
apparent,  as  PP  drops  rapidly  during  the  stroke,  and  when  sin  6  is  maxi- 
mum, PP  is  much  less.  A  few  trials  will  locate  the  angle  at  which  PP 
sin  0  is  maximum. 

More  correctly,  the  value  pA  given  by  Formula  (53),  Chap.  XVI  gives 


566  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

the  pressure  on  the  guide,  but  as  PN  is  apt  to  give  a  greater  value  it  is 
usually  safe.  For  horizontal  engines  the  weight  of  the  crosshead  and  a 
portion  of  the  connecting  rod  and  piston  rod  should  be  added  to  pA. 
This  may  result  in  a  value  greater  than  PN  but  in  view  of  the  arbitrary 
values  for  allowable  pressure  Ps,  varying  greatly  as  given  by  different 
authorities,  too  great  refinement  seems  out  of  place.  PN,  Px  and  p 
should  properly  include  the  inertia  of  the  reciprocating  parts  (not  includ- 
ing the  connecting  rod),  but  this  is  often  neglected  in  strength  calcu- 
lations (see  Chap.  XVI,  and  Chap.  XXI,  Par.  166). 

As  a  suggestion  in  proportioning  crosshead  shoes  for  steam  engines, 
the  author  has  used  the  following  empirical  formula  for  the  length  of 
shoes  for  a  number  of  years  in  Corliss  engine  design : 

L  =  0.75D  +  5  (10) 

This  is  in  inches.  The  width  w  was  taken  one-half  of  the  length.  The 
area  would  be  1.5L,  and  the  bearing  pressure  per  sq.  in.  on  any  shoe  is: 


AS      wL 

This  also  applies  to  trunk  pistons  which  serve  as  crossheads;  w  is  then 
replaced  by  D. 

186.  Application  of  Formulas. — In  designing  the  pin  for  a  steam  engine, 
the  bearing  pressure  is  usually  assumed.  Let  a  pin  be  designed  for  the 
standard  20-in.  Corliss  engine  of  Chap.  XII,  Par.  64.  The  steam  pres- 
sure p  is  125  lb.  Let  PI  =  1200  and  P  =  1000  (for  the  crank  pin). 
Then  from  (3) :  

/   IOC 

I  =  0.887  X  20  J^  =  6.25  in. 


The  nearest  safe  pin  in  Table  89  is  6J^  in.  long,  and  the  diameter  is  5J£  in. 
and  this  will  be  used.  Assume  b  =  I  in  Table  89  (Par.  187).  Taking  10 
-  1.51  and  q  =  J£,  (6)  gives: 

s  =  2  X  125  X400  X  6.5  =  ^  ?#$$& 

Neglecting  the  angularity  of  the  connecting  rod,  a  %  cut-off  occurs 
when  the  crank  is  at  point  8,  Table  57,  Chap.  XVI.  The  value  of  FP  (in- 
ertia of  piston,  piston  rod  and  crosshead)  at  this  point  is  6450  Ib. 
Added  to  39,250  (being  Px)  gives  45,700  Ib.  The  value  of  pA  from  the 
same  table  is  1306  Ib.  The  resultant  is: 


PR  =  V45,7002  +  13062  =  46,200  Ib. 
Then: 

_     46,200 
^  -  5.5  X  6.5 


CROSSHEADS  567 

This  is  but  slightly  larger  than  1200,  as  the  pin  diameter  was  increased 
to  5J^  in.,  but  had  I  been  taken  so  that  PI  was  exactly  1200  Ib.  for  steam 
pressure  only,  the  ratio  of  actual  to  assumed  pressure  on  the  pin  would 
be: 

P«  _  46,200 
Px      39,250 
The  actual  stress  is: 

P*       3920  X  46,200 


\  39,250 

The  standard  factor  of  safety  for  reversed  stress  in  Par.  166,  Chap. 
XXI  is  6;  taking  an  elastic  limit  of  38,000  as  given  in  Table  73,  Chap.  XXL 
the  factor  of  judgment  is: 

38,000 

**      6  X  4620 
This  shows  that  the  pin  is  safe. 

For  a  Diesel  engine  assume  the  pin  length  to  be  0.55D.  Assume  a 
10  in.  cylinder  and  a  pressure  of  500  Ib.  Then  10  =  8.25.  The  factor  of 
safety  from  Table  82,  Chap.  XXI  is  3.  If  the  factor  of  judgment  is 
1.25,  total  factor  of  safety  is  3.75;  calling  this  3.8  gives  an  allowable 
working  stress  of  10,000  Ib.  Taking  q  =  0.5  as  before,  (7)  gives: 

,       3  /2  X  500  X  100  X  5.5       Q  .  07/  . 

d==\-  10,000  =3.81;  say  3%  m. 

The  bearing  pressure  per  sq.  in.  of  projected  area  is: 
500  X  78.54 


which  is  practically  the  value  allowed  in  Table  19,  Chap.  XI.  Inertia 
would  reduce  this  some  after  the  engine  is  started. 

In  detailing  the  piston  and  connecting  rod  end  it  might  be  found  that 
slightly  different  proportions  would  give  better  results,  but  the  size  of 
pin  found,  3%  by  5J£  in.,  seems  reasonable. 

Designing  the  crosshead  shoe  for  the  20  in.  Corliss  engine,  with  a  rod 
6  cranks  long  gives,  from  (9). 

0.7854  X  202  X  125 
PN  = ,    ,        =  6630. 


From  (10): 

L  =  15  +  5  =  20. 
From  (11): 


-  568 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


The  maximum  value  of  pA  from  Table  57,  Chap.  XVI  is  5432  lb.,  being 
less  than  PN.  Adding  the  weight  of  the  crosshead  and  part  of  the  con- 
necting rod  and  piston  rod  brings  this  up  to  7000  lb.  This  gives: 

7000 


Ps  = 


200 


=  35  lb. 


The  difference  is  negligible.     The  result  is  still  less  than  the  allowable 
pressure  given  in  Table  19,  Chap.  XI. 

In  the  internal-combustion  engine  the  maximum  guide  pressure  occurs 
earlier  in  the  stroke.  In  the  automobile  engine,  the  data  for  which  is 
given  in  Chap.  XVI,  the  maximum  pressure  from  (8)  occurs  at  crank 
position  3,  and  PP  is  31.8  lb.  per  sq.  in.  including  the  effect  of  piston  inertia. 
The  cylinder  being  3^  in.  in  diameter,  the  total  maximum  guide  pressure 


/ft  §.  9 ,  use  2  oil  holes 
as  on  crank  pin 


Crosshead  Pin 


FIG.  391.' 


is  264  lb.  The  piston  is  4  in.  long,  making  a  projected  area  of  13  sq.  in. 
This  gives  a  pressure  of  20.3  lb.  per  sq.  in.  The  rings  extend  about  t  inch 
from  the  head  end.  If  this  portion  is  deducted  the  unit  pressure  is  27  lb. 
The  actual  unit  pressure  is  somewhere  between  these  two  values.  In 
Table  19,  Chap.  XI,  the  allowable  pressure  for  a  trunk  piston  in  an 
internal  combustion  engine  is  21  lb.  As  economy  of  space  and  extreme 
lightness  are  requirements  of  an  automobile  engine,  higher  pressures  are 
more  likely  to  be  used  than  in  engines  for  industrial  service. 

187.  Designs  From  Practice.  —  Fig.  391  shows  a  crosshead  pin  used  by 
the  author  on  Corliss  engine  work,  and  Table  89  gives  data  for  pins  from 
3  to  8J^  in.  in  diameter.  The  notation  applies  only  to  the  table.  These 
pins  were  designed  from  Formulas  (3)  and  (4),  taking  p  =  125,  P  =  1000 
and  PI  =  1200.  It  was  Ttlso  assumed  that  b  =  L  Crosshead  pins  for 
standard  steam  engines  for  the  pressures  just  named  are  included  with 
crank  pins  in  Table  91,  Chap.  XXVII. 


CROSSHEAD  8 


569 


Figure  392  is  one  form  of  crosshead  used  on  the  Bass  Corliss  engine, 
built  by  the  Bass  Foundry  and  Machine  Co.,  Fort  Wayne,  I nd.  The 
shoe  adjustment  is  such  that  the  adjusting  nut  draws  in  the  direction  the 


shoe  slides,  preventing  a  bending  strain  on  the  bolt.  It  is  a  turned 
crosshead,  sliding  in  bored  guides.  Th'e  body  of  the  crosshead  is  a  steel 
casting  and  the.  shoes  cast  iron,  babbitted. 

Figure  393  is  the  Mclntosh  and  Seymour  crosshead,  used  on  Type  F 


570  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  89 


d.in. 


Z.in. 


If  6  =  I 


d\,  in. 


dz,  in. 


da,  in. 


<,  in. 


3^6 

4 

4% 
4% 


9 
10 


We 


2.60 
3.33 
3% 
4.02 

4.91 

5.79 
6.27 

7.64 


1 
1 

i-H 


i 

1H 


^^ 


H 


FIG.  393. — Mclntosh  and  Seymour  crosshead. 

steam  engine  biiilt  by  the  Mclntosh  and  Seymour  Corporation,  Auburn, 
N.  Y.    The  pin  is  flattened  top  and  bottom,  and  the  brasses  of  the  connect- 


CROSSHEADS  571 

ing  rod  boxes  are  cut  away  so  that  there  is  a  small  wipe-over  in  the  posi- 
tion of  extreme  angularity  of  the  rod.  The  pin  is  held  in  place  by  three 
tap  bolts.  When  it. is  desired  to  withdraw  the  pin,  these  bolts  are  re- 
moved and  the  pin  forced  out  by  two  set  screws  in  the  pin  flange. 

The  body  of  the  crosshead  is  a  steel  casting  and  the  shoes  are  cast  iron 
faced  with  babbitt  metal.  The  shoes  have  no  adjusting  wedges,  but 
shims  or  liners  may  be  used  in  case  of  wear. 


CHAPTER  XXVII 
CRANKS 

188.  Introduction. — In  this  chapter  it  is  intended  to  cover  the  over- 
hung crank.  This  is  usually  a  casting  of  iron  or  steel  forced  on  the 
shaft,  and  usually  the  crank  pin  is  forced  in.  When  pin  and  shaft 
diameters  are  large  relative  to  the  stroke,  a  steel  casting  with  crank  and 
pin  cast  integral  is  sometimes  used;  this  is  then  forced  onto  the  shaft. 

The  problem  is  to  obtain  proper  bearing  surface  for  the  pin,  ample 
strength  and  stiffness  of  pin  and  crank,  and  the  securing  of  the  pin  and 
shaft  to  the  crank. 

The  center  crank  and  multi-cylinder  crank  will  be  treated  under 
shafts  in  Chap.  XXVIII. 

In  general,  the  over-hung  crank  is  used  only  for  steam  engines  and 
large  double-acting  gas  engines.  The  factor  of  safety  (/A/T)  in  Table 
82  of  Chap.  XXI  for  these  may  be  taken  as  5  and  4.2  respectively,  for 
ductile  materials.  Going  on  the  assumption  of  Par.  159,  Chap.  XXI, 
that  for  reversed  stress  the  compressive  stress  of  cast  iron  may  be  taken 
as  one-fifth  of  its  actual  value,  fA)  the  standard  factor  of  Chap.  XXI  may 
be  taken  as  7.2  for  both  steam  and  gas  engines  of  this  type.  The  prod- 
uct /4/r  will  then  be  7  for  the  steam  engine  and  5  for  the  gas  engine. 
A  factor  of  judgment  may  be  used  in  addition  to  these  if  desired.  The 
value  7  was  taken  for  steam  as  the  value  fA  for  ductile  materials  in 
Table  82,  Chap.  XXI  is  5 — not  a  completely  reversed  factor. 

Inertia  and  variation  of  pressure  is  provided  for  by  these  factors,  as 
explained  in  Par.  166,  Chap.  XXI,  so  in  applying  them,  the  force  acting 
is  assumed  to  be  that  produced  by  maximum  unbalanced  steam  or  gas 
pressure  only. 

Notation. 

d  =  diameter  of  crank  pin  (in  bearing)  in  inches. 
di  =  diameter  of  crank  pin  fit. 

dH  =  outside  diameter  of  pin  or  shaft  hub,  in  inches. 
I  =  length  of   crank  pin  bearing  in  inches. 

10  =  length  of  moment  arm  for  computing  strength  of  pin  or  crank 
arm,  in  inches. 

572 


CRANKS  573 

D  =  diameter  of  cylinder  in  inches. 
Ds  =  diameter  of  standard  cylinder  when  some  standard  pressure  is 

assumed,  as  explained  in  Par.  63,  Chap.  XII. 
t  =  thickness  of  hub  considered  as  a  thick  cylinder. 
b  =  effective  breadth  of  crank  arm  in  inches,  as  shown  on  Fig.  396. 
h  =  depth  of  arm  section  in  inches. 
A  =  area  of  arm  section  in  square  inches. 
/  =  moment  of  inertia  of  arm  section. 
c  =  distance  from  neutral  axis  to  extreme  fiber,  in  inches. 
S  =  stress  in  pounds  per  square  inch. 
SB  =  bending  stress  in  arm. 
SD  =  direct  stress  in  arm. 
Sc  =  crushing  stress  in  key. 
px  =  total    maximum   unbalanced  pressure  on  piston  in  pounds, 

due  to  steam  or  gas  pressure  only. 

PT  =  turning  effort  at  crank  pin  (tangential)  in  pounds. 
PP  =  total  unbalanced  pressure  at  any  part  of  stroke  in  pounds,  in- 
cluding inertia. 
P  =  maximum  allowable  pressure  on  pin  per  square  inch  of  projected 

area. 

PM  =  mean  pressure  per  square  inch  of  piston  area  per  cycle,  in- 
cluding  inertia   of  reciprocating   parts  and  connecting  rod. 
This  must  be  taken  from  a  complete  stroke  diagram  for  the 
cycle.     PM  may  be  found  from  diagrams  of  Par.  105,  Chap. 
XVI,  both  in  magnitude  and  direction  (in  line  of  stroke  only). 
p  =  maximum  unbalanced  pressure  per  sq.  in.  in  cylinder. 
T  =  tons  per  in.  of  diameter  per  in.  of  length  required  to  complete 

pressed  fit. 

N  =  r.p.m.  of  engine  shaft. 
k  =  l/d. 

C  =  a  constant  in  wear  formula, 
jx  =  coefficient  of  friction. 

189.  Crank  Pin. — The  over-hung  crank  pin  may  be  designed  inde- 
pendent of  the  shaft  as  it  transmits  no  power  from  one  part  of  the  shaft 
to  another.  Fig.  394  shows  two  methods  of  pin  design.  In  Fig.  394A 
the  pin  is  reduced  from  %  to  Y±  in.  in  diameter — depending  upon  the  size — 
where  it  enters  the  crank.  In  Fig.  3945  a  collar  is  turned  on  the  pin  and 
the  portion  in  the  crank  may  be  equal  or  greater  in  diameter  than  the 
wearing  part.  In  this  pin  the  moment  arm  10  is  greater  than  one-half 
the  bearing  length  L  Should  the  diameter  d  be  much  less  than  the 


574 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


diameter  of  fit  in  the  crank,  it  should  be  checked  for  strength  by  using 
a  moment  arm  1/2. 

A  pin  may  be  designed  for  strength  and  checked  for  bearing  pressure 
and  wear;  or  it  may  be  designed  for  bearing  pressure  and  checked  for 
wear  and  strength.  The  latter  method  will  be  used  here. 

As  the  method  of  assuming  allowable  bearing  pressures  is  usually 
based  upon  direct  piston  thrust,  the  additional  load  on  the  pin  due  to 
angularity  of  the  connecting  rod  may  be  neglected,  greatly  simplifying 
calculations.  Let  Px  be  the  total  maximum  unbalanced  pressure  on  the 
piston,  P  the  pressure  per  sq.  in.  of  projected  area  of  the  pin,  and  A  the 
projected  area  of  the  pin  journal  in  sq.  in.  The  general  equation  is  then: 


Px  =  PA 


(1) 


r 


1 


i-    - 


FIG.  394. 

If  p  is  the  maximum  unbalanced  pressure  per  sq.  in.  on  the  piston 


also 

Substituting  in  (1)  gives: 


4 
A  =  dl: 


4P 


Let  I  =  kd,  then : 


D 


(2) 
(3) 


The  value  of  P  may  be  selected  from  Table  19,  Chap.  XI,  and  k  may 
be  assumed,  or  it  may  be  calculated  from  (4),  Chap.  XI;  this  formula, 
however,  must  be  used  with  judgment,  and  it  is  better  used  as  a  check 
after  selecting  some  value  of  k  found  to  be  satisfactory  in  practice. 


CRANKS  575 

The  pin  may  now  be  checked  for  strength.     The  modulus  of  section 
of  a  circular  section  in  bending  is: 


32 

Then  if  S  is  the  stress  : 

irdi'-S 
-33-    = 

From  which,  combining  with  (1)  : 

Z2PX10  .0 

o  =  -  r~r~  =  -  r~*  —  (4) 

irdi3  di3 

It  may  safely  be  assumed  that  di  and  d  are  equal;  then  the  value 
found  may  be  taken  as  the  smaller  of  the  two,  and  (4)  may  be  written: 


It  is  usually  assumed  that  the  load  Px  acts  at  the  center  of  the  pin; 
this  would  be  theoretically  correct  if  it  were  taken  either  as  a  concentrated 
or  distributed  load.  For  Fig.  394A,  10  would  equal  Z/2;  for  Fig.  394£, 
the  thickness  of  the  collar  must  be  added  to  this. 

Except  in  locomotive  practice,  Fig.  394A  is  the  most  usual  design; 
then  10  =  1/2,  and  a  special  formula  for  this  becomes  : 


8  =  =  5.1  WP  (6) 

It  is  good  practice,  and  now  quite  common,  to  have  k  equal  to  unity;. 
then  (6)  becomes: 

S  =  5.1  P  (7) 

This  makes  a  good  rigid  pin  with  small  deflection. 

Perhaps  the  most  direct  method  of  determining  the  pin  dimensions 
is  as  follows.     Formula  (6)  may  be  written  : 


The  maximum  pressure  P  may  be  taken  from  Table  19,  Chap.  XI  (pM 
in  this  table),  and  S  may  be  determined  by  using  the  factor  of  safety 
given  in  the  introduction.  Assuming  the  elastic  limit  as  38,000,  these 
data,  with  the  resulting  values  of  k  are  given  in  Table  90  for  two  values 
of  the  factor  of  judgment  /3  (see  Chap.  XXI,  Par.  166). 

Selecting  k  with  the  corresponding  value  of  P  from  Table  90,  (3)  will 
give  the  diameter;  the  length  is  obviously  equal  to  kd.  These  dimensions 
are  usually  rounded  up  to  eighths,  quarters  or  even  halves  of  an  inch  in 
the  larger  sizes,  and  if  desired  may  be  checked  for  new  values  of  S  and 
P,  but  this  is  seldom  necessary. 


576 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Steam  Engine. — Assuming  k  as  unity,  Table  91  has  been  computed  for 
steam  engine  crank  pins  of  the  style  shown  in  Fig.  394A.  It  is  assumed 
that  the  maximum  unbalanced  pressure  p  is  125,  and  that  P  is  1000  Ib. 


TABLE  90 


Steam  engines 

Internal-combustion  engines 

/3 

/  . 

8 

p 

A; 

/ 

s 

P 

K 

1.00 

5.00 

7600 

1000 
1200 

1.22 
1.12 

4.20 

9050 

1400 
1700 

1.13 
1.02 

1.25 

6.25 

6100 

1000 
1200 

1.10 
1.00 

5.25 

7230 

1400 
1700 

1.01 
0.92 

TABLE  91 


Ds,  in. 

Crosshead  pin 

Crank  pin 

d,  in. 

I,  in. 

d,  in.                                I,  in. 

10 

3 

3H 

$H 

3H 

12 

3K 

4 

4 

4 

14 

4 

4M 

•4H 

4M 

16 

V6 

5>i 

Vi 

5}£ 

18 

5 

6 

6 

6 

20 

5K 

*x 

6H 

&y2 

22 

6 

iy* 

7M 

7M 

24 

W 

8 

8 

8 

26 

7 

8H 

8>i 

8>i 

28 

7H 

9 

9 

9 

30 

8H 

10   . 

10 

10 

The  pressure  p  ( =  125)  may  be  considered  the  standard  pressure  acting 
in  standard  cylinders  as  explained  in  Par.  63,  Chap.  XII. 

From  (3),  d  =  Q.313DS.  The  dimensions  of  the  crosshead  pin  are 
also  given,  based  on  a  pressure  of  1200  Ib.  per  sq.  in.  of  projected  area. 
The  length  is  taken  equal  to  that  of  the  pin,  as  explained  in  Chap. 
XXVI;  the  diameter  is  then  %  the  diameter  of  the  crank  pin. 

Figure  395  and  Table  92  are  for  standard  crank  pins  used  by  the  author 
for  Corliss  engines.  The  notation  only  applies  to  the  figure  and  table 
should  it  disagree  with  that  used  elsewhere  in  this  chapter. 

While  these  tables  were  used  for  several  years  on  successful  engines, 
they  are  given  mainly  to  illustrate  convenient  methods  of  arranging  de- 
sign data.  The  selection  of  dimensions  for  other  steam  pressures  is  ex- 
plained in  Par.  63;  Chap.  XII. 


CRANKS 
TABLE  92 


577 


d,  in. 

di,  in. 

6,  in. 

dz,  in. 

2i,  in. 

c,  in. 

ds,  in. 

Oil  holes 

3K 

4% 

^ 

1 

IK 

IK 

K 

i-K" 

4 

5K 

% 

1 

IK 

IK 

H 

i-K'r 

4K 

5% 

K 

IK 

1M 

l^ 

% 

i-K" 

5H 

7K 

H 

IK 

1% 

2 

% 

i-K" 

6 

7% 

i 

IK 

1% 

2K 

H 

i-%r/ 

6K 

8K 

IK 

1% 

2K 

2^ 

« 

i-%- 

•   7K 

9% 

IK 

IH 

2K 

2^ 

^ 

i-%" 

8 

10H 

IK 

IK 

2K 

3 

H 

i-K" 

8K 

11 

iH 

1« 

2K 

3K 

K 

i-K" 

9 

n*i 

IK 

1% 

2K 

3% 

K 

2-%" 

10 

13 

1% 

i« 

2^ 

3^ 

K 

2-%" 

The  method  of  finding  the  mean  pressure  per  sq.  in.  of  piston  area, 
PM  acting  on  the  pin  is  described  in  Par.  104,  Chap.  XVI.     For  steam 


Shaft  Side  ofP/n 
on  Dead  Center 


FIG.  395. 

engines  this  is  practically  the  m.e.p.  For  engines  with  long-range 
cut-off  it  approximates  the  maximum  pressure  in  the  cylinder  and  may 
be  safely  taken  as  such. 

Adopting  the  notation  of  this  chapter,  Formula  (1),  Chap.  XI  may 
be  written: 

T\9r>     jAr 

(9) 


The  value  of  C  is  taken  from  Table  93  of  the  same  chapter.     Taking  this 
value  and  solving  for  JV,  the  corresponding  rotative  speed  may  be  found. 

37 


578 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


With  p  =  PM  =  125,  P  =  1000  and  k  =  1;  the  values  used  for  Table 
91  substituted  in  (3)  and  the  value  of  d  so  found  substituted  in  (9),  we 
have: 

_  4.3  C    fkp  _  2430 
Ds 


•L V       T-»        -rxA  /     T-k       


(10) 


TABLE  93 


Di,  in. 

N 

Di,  in. 

N 

Di,  in. 

.V 

2 

1220 

12 

202 

32 

76 

3 

810 

14 

174 

34 

71 

4 

610 

16 

152 

36 

67 

5 

487 

18 

135 

38 

64 

6 

405 

20 

122 

40 

61 

7 

347 

22 

110 

42 

58 

8 

304 

24 

101 

44 

55 

9 

270 

26 

93 

46 

53 

10 

243 

28 

87 

48 

50 

11 

221 

30 

81 

50 

48 

It  is  interesting  to  note  these  results  in  Table  93,  which  compare  favor- 
ably with  conservative  speeds  often  used.  Higher  speeds  than  those  given 
in  the  table  have  been  used  for  large  engines,  but  it  is  stated  in  Par.  52, 
Chap.  XI,  that  the  value  of  C,  which  was  taken  as  200,000,  may  be 
doubled  with  excellent  design  and  conditions  of  operation. 

Internal-combustion  Engine. — For  these,  the  mean  pressure  on  the 
pin  must  be  used  for  PM  in  (10),  and  this  depends  upon  the  indicator 
and  inertia  diagrams  as  already  explained.  Gtildner  assumes  for  his 
standard  diagram  128  Ib.  for  the  expansion  stroke,  18  Ib.  for  each  idle 
stroke  (exhaust  and  suction)  and  32  Ib.  for  the  compression  stroke. 
The  mean  of  these  is  49  Ib.  The  value  of  C  from  Table  19,  Chap.  XI 
is  90,000.  Taking  k  =  1,  P  =  1400  and  p  =  400,  (10)  becomes: 

"      4220 


Ds 

Then  from  (3)  for  the  same  assumptions: 

d  =  OA73DS 


(12) 


Guldner  takes  S  as  12,000.  For  the  factor  of  safety  already  used  Se 
must  be  50,000  if  /3  =  1,  or  63,000  if  f3  =  1.25.  With  other  data  as  be- 
fore, this  gives,  from  (8) : 


rs 

=  W~: 


,000 


X  1400 


=  1.3  nearly. 


CRANKS 


579 


and  from  (3) : 

d  =  QA15Da  (13) 

This  increases  the  allowable  value  of  A7"  by  14  per  cent. 

From  (6),  the  value  of  S  when  d  is  as  given  by  (12)  is  7150 — practically 
the  same  as  in  Table  90  when /3  is  1.25.  The  best  value  of  S  may  be 
determined  only  when  the  material  is  known.  It  is  better  to  take  the 
value  of  PM  from  diagrams  drawn  for  a  particular  design — at  least  for  a 
particular  type. 

By  using  a  greater  value  of  P  the  ratio  k  is  reduced,  decreasing  the 
moment  arm  for  the  crank  and  shaft.  If  the  product  kP  is  constant,  (3) 
shows  that  the  pin  diameter  is  not  changed.  It  may  also  be  seen  from  (8) 


FIG.  396. 

that  this  makes  PS  constant,  and  these  relations  may  be  convenient  in 
determining  conditions. 

In  all  pin  design  there  should  be  a  fillet  where  the  pin  changes  diame- 
ter. If  the  pin  is  cast  integral  with  the  crank  there  should  be  a  fillet 
where  they  join. 

Crank  Arm. — A  crank  arm  of  substantial  design  is  shown  in  Fig. 
396.  This  is  a  plain  crank  having  all  the  requisites  to  transmit  power  from 
the  crank  pin  to  the  shaft.  Sometimes  a  counterbalance  is  added,  and 
sometimes  a  disc  crank  is  made  by  adding  rim  and  web,  but  these  should 
not  be  depended  upon  to  add  to  the  strength  of  the  crank  and  may  be 
omitted  for  the  present  discussion. 

In  a  crank  of  this  general  design  the  greatest  liability  to  failure  is 


580  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

through  the  pin  hole  when  the  crank  is  on  one  of  the  dead  centers,  and 
this  will  first  be  considered. 

Assuming  a  perfectly  rectangular  section,  the  bending  stress  is: 


The  direct  stress  is: 

SD  =  ±6f 

These  act  at  the  same  time  at  the  dead  center,  PP  being  the  total  force  in- 
cluding steam  or  gas  pressure  and  all  inertia  effects.  The  sign  is  plus 
for  tensile  stress  and  minus  for  compressive.  The  total  stress  is  : 


At  the  beginning  of  the  stroke  both  signs  are  minus  on  the  pin  side, 
or  face  of  the  crank,  and  for  the  back  side  the  first  sign  is  plus  and  the 
second,  minus.  At  the  end  of  the  stroke  both  signs  are  plus  on  the  face, 
while  on  the  back  the  first  is  minus  and  the  second  plus. 

From  the  equations  of  Par.  166,  Chap.  XXI,  the  worst  condition  for 
given  indicator  and  inertia  diagrams  may  be  determined. 

Cast  Iron  Crank.  —  For  the  conditions  assumed  in  Table  82,  Chap.  XXI, 
for  the  factor  5  for  steam  engine  cranks  there  is  nearly  full  reversal,  and 
the  factor  7  may  be  assumed  for  cast  iron  as  previously  explained.  There 
is  an  initial  stress  in  the  metal  around  the  pin  presumably  higher  than  the 
liveload  stress.  The  factor  for  this  may  be  4  as  for  static  stress.  Due  to 
this  initial  stress  and  the  fact  that  the  conditions  assumed  in  determining 
the  factor  7  only  obtain  for  very  large  overloads;  also  because  the  result- 
ing dimensions  are  greater  than  those  of  many  actual  cranks,  the  factor 
of  safety  may  be  taken  as  5.  This  factor  will  be  applied  by  making  PP 
in  (14)  equal  to  Px  and  taking  both  signs  positive.  Then  the  general 
formula  is: 


The  diagrams  of  Fig.  186,  Chap.  XVI,  drawn  to  scale  for  the  20  in. 
Corliss  engine  previously  referred  to,  when  analyzed  by  the  method  of 
Par.  166,  Chap.  XXI  give  a  factor  of  4.3  for  %  cut-off.  For  %  cut-off 
the  factor  is  5.55.  These  are  both  for  cast  iron.  Then  in  view  of  the 
initial  stress  in  the  crank-pin  eye  due  to  forced  fit,  the  factor  5  seems 
safe.  It  is  likely  that  the  iron  used  for  cranks  would  usually  have  a 
tensile  strength  greater  than  16,000  lb.,  but  this  value  will  be  used  here. 

An  old  rule  was  to  make  the  length  of  crank  fit  %  of  the  shaft  diame- 


CRANKS  581 

ter.  Another  rule  was  to  make  the  diameter  of  the  shaft  ^  the  cylinder 
diameter.  This  makes  the  length  of  the  crank  %  the  cylinder  diameter. 
This  may  be  done  satisfactorily  in  steam  engine  practice  by  taking  the 
cylinder  diameter  as  that  of  the  standard  cylinder  Ds  carrying  some  stand- 
ard pressure.  For  any  other  pressure  (and  this  may  be  used  for  internal- 
combustion  engines  if  desired)  the  length  of  the  fit  may  be  found  as  for 
other  standard  dimensions  as  explained  in  Par.  63,  Chap.  XII,  or  for 
compound  engines  in  Par.  72,  Chap.  XIII. 

If  h,  the  arm  depth,  is  made  0.32Z)(S,  there  is  room  for  an  oil  groove  on 
the  hub,  even  on  the  smallest  sizes.  Then  taking  Px  as  referred  to  the 
standard  cylinder  Ds,  and  substituting  the  value  of  h  in  (15),  we  have: 


which  is  a  special  formula  when  h  =  0.32D,s. 

As  it  is  well  to  confine  the  use  of  cast  iron  cranks  to  steam  engines  not 
carrying  high  pressures,  another  special  formula  may  be  given,  based  upon 
the  pin  dimensions  of  Table  91.  Then  let  p  =  125,  S  =  3200  and  10 
=  0.32As,  for  which: 

6  =  0.672£>s  =  2.ld  (17) 

This  dimension  may  be  obtained  as  shown  in  Fig.  396,  and  while  rupture 
might  not  occur  in  line  with  the  pin  diameter,  the  method  has  proven 
satisfactory  in  practice. 

Steel  Cranks.  —  For  steam  engines  carrying  high  pressures  and  for  gas 
engines,  cranks  should  be  steel  castings  As  dimensions  are  not  so  exces- 
sive there  is  no  difficulty  in  applying  the  formula  with  the  factors  already 
assumed  in  the  introduction,  and  used  for  the  pin.  Formulas  (14)  to 
(16)  also  apply  to  steel  cranks 

From  Table  73,  Chap.  XXI,  the  elastic  limit  is  given  as  30,000  Ib. 
Taking  S  as  6000  and  other  data  as  before,  a  special  formula  for  steam 
engines  is: 

b  =  0.36As  =  I.12d  (18) 

For  gas  engines,  calculations  show  a  massive  construction  when  the 
pin  is  to  be  forced  in.  It  is  no  doubt  the  best  practice  to  cast  the  pin 
with  the  crank.  As  an  example,  the  proportions  of  (12)  will  be  assumed 
in  which  k  =  1.  Let  a  standard  pressure  p  be  400;  then  S  is  7150.  Let 
it  be  assumed  that  the  length  of  the  hub  fit  is  0.5DS  and  that  h  =  QA2DS. 
Then  as  d  =  QA73DS,  10  may  be  taken  as  QA5DS.  Substituting  these  in 
(16)  gives: 

b  =  1.3DS  =  2.7d  (19) 

which  is  excessive. 


f 

x-4— (— 

_ 


582  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  crank  arm  is  sometimes  cut  away  as  shown  in  Fig.  398.  This  is 
permissible,  especially  in  steel  castings,  but  is  not  logical  design.  It 
gives  a  poor  section  for  torsional  stiffness  when  the  crank  center  is  normal 
to  line  of  stroke;  it  also  necessitates  joining  a  light  and  heavy  section  if  the 
walls  are  made  thin.  It  may  lighten  the  crank  if  the  stroke  is  long,  and 
it  reduces  the  moment  arm  by  moving  the  neutral  axis  toward  the  pin ; 
but  it  also  reduces  the  section  modulus  and  should  be  carefully  checked 
for  tension  on  the  side  away  from  the  pin.  To  find  the  stress  in  this 
section  the  general  formula  is : 

S--=PX10  (20) 

c 

where  7  is  the  moment  of  inertia  about  axis 
XX  and  c  is  the  distance  to  the  extreme  fiber 
as  shown  in  Fig.  397. 

Well  designed  cranks  seldom  need  check- 
FlG<  397'  ing   when   in   a  position  normal  to  line  of 

stroke.  They  are  then  subjected  to  torsion  and  bending.  The  bending 
moment  is: 

MB  =  PPlx 

where  lx  is  the  distance  from  the  pin  center  to  any  section  as  shown  in 
Fig.  396.  The  modulus  of  section  for  this  case  is  about  axis  YY.  The 
twisting  moment  is : 

M  T  =  Pplo 

where  10  is  taken  from  Fig.  396.  If  the  section  is  rectangular  the  maximum 
shearing  stress  is  at  the  center  of  the  surface  of  the  long  side  and  may  be 
found  from  Par.  163,  Chap.  XXI.  At  the  center  of  the  short  side  the 
stress  is  proportional  inversely  to  the  stress  at  the  long  side,  as  the  lengths 
of  the  sides  of  the  rectangle.  These  stresses  may  be  combined  by  (20) , 
Chap.  XXI.  As  stated  in  Par.  166,  Chap.  XXI,  the  factor  of  safety  for 
this  position  may  be  taken  as  3  for  ductile  materials  and  6  for  brittle 
material,  such  as  cast  iron. 

190.  Hubs. — The  subject  of  pressed  fits  is  treated  in  Par.  178,  Chap. 
XXIV,  so  the  discussion  will  not  be  repeated. 

Let  T  be  tons  per  inch  of  diameter  per  inch  of  length  required  to  com- 
plete the  fit;  let  /i  be  the  coefficient  of  friction  and  t  the  thickness  of  hub. 
Then  from  (28),  Chap.  XXIV: 


and  from  (29),  same  chapter: 


CRANKS 


The  outside  diameter  is  then: 


da  =  d  +  2t 


583 


(23) 


For  cast  iron  cranks,  taking  /*  =  0.25,  T  =  0.8  and  S  =  4000,  as- 
suming static  loading: 

q  =  0.51 
and 

t  =  0.5d. 
Then: 

dH  =  2d  (24) 

The  allowable  static  stress  in  a  steel  casting  is  15,000  Ib.  If  T  is 
1.25,  the  outside  diameter  could  be  less  than  1.3d.  If  the  stress  is  taken 
at  6250  when  T  is  1.25,  d  =  2d  as  just  found  for  cast  iron.  There  are 


FIG.  398. — Nordberg  crank  and  pin. 

other  considerations  besides  strength  in  crank  design,  such  as  the  necessity 
of  a  diameter  sufficient  to  cover  the  bearing,  but  the  formulas  may  be 
used  as  a  check. 

Crank  pins  are  forced  in  and  riveted  over  cold  as  shown  in  Fig.  396. 
There  is  no  tendency  for  the  pin  to  turn  so  no  key  is  needed,  but  as  all  the 
power  of  the  engine  is  transmitted  through  the  shaft  fit,  the  shaft  is 
keyed  to  the  crank  with  one  or  two  keys  as  a  precaution  against  slipping. 

In  Fig.  396,  if  the  fit  were  not  tight  enough  to  keep  the  crank  from  slip- 
ping on  the  shaft,  the  surface  ma  of  the  keyway  must  sustain  the  entire 
load.  If  Sc  is  the  crushing  stress  in  the  key,  the  moment  of  resistance  of 
the  key  is  approximately. 

d0 


584 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


This  must  equal  the  twisting  moment  FT?,  where  r  is  the  radius  of  the 
crank  circle  in  inches.     Then: 


Should  the  key  take  the  entire  load  the  standard  factor  of  safety  would 
be  3  for  repeated  load.  It  is  probable  in  most  cases  that  the  fit  would 
hold  without  a  key,  so  if  desired,  a  factor  of  judgment  less  than  unity  may 
be  applied. 

For  steam  engines  the  maximum  value  of  PT  may  be  taken  as  Px- 
For  internal-combustion  engines,  Giildner  finds  that  the  maximum  PT  is 
0.5P*  for  his  standard  reference  diagram. 


FIG.  399. — Bass-Corliss  disc  crank. 


FIG.  400. — Plain  crank  with  counterbalance. 


By  assuming  values  in  terms  of  Ds,  a  special  formula  may  be  derived. 
In  doing  so  it  is  best  to  retain  d0  and  >SC. 

For  the  steam  engine,  as  already  assumed,  a  =  Q.375DS.  Also  let 
r  =  l.2Ds  which  covers  most  cases.  Assume  p  as  125.  Then: 


m  =  —, 


d0  Sc 


(26) 


For  the  internal-combustion  engine  we  may  take:  a  =  0.5D,  r  = 
0.8Z)sandp  =  400.     Then: 

(27) 


m 


These  two  special  formulas  may  be  used  for  preliminary  calculations. 

If  two  keys  are  used  they  are  placed  90  degrees  apart.     The  value  of  m 

for  each  key  may  be  taken  as  one-half  that  for  a  single  key,  or  perhaps 


CRANKS 


585 


some  greater.  This  arrangement  weakens  the  shaft  less  than  a  single 
key. 

The  factor  of  judgment  may  be  taken  as  0.75,  making  a  total  factor  of 
2.25,  which,  with  an  elastic  limit  of  38,000  gives  Sc  =  17,000. 

191.  Crank  Designs. — Fig.  398  shows  a  crank  used  on  steam  engines 
built  by  the  Nordberg  Manufacturing  Co.,  Milwaukee,  Wis.  The  hub 
projection  next  to  the  bearing  is  omitted,  except  that  necessary  for  facing. 
A  small  amount  may  be  placed  on  the  opposite  side  so  long  as  it  does  not 
interfere  with  the  connecting  rod.  This  arrangement  either  shortens 
the  moment  arm  used  in  determining  the  shaft  strength,  or  makes  the 
crank  arm  thicker,  or  both.  A  thicker  arm  reduces  the  difficulties  en- 
countered in  determining  b. 


FIG.  401. — Gas  engine  crank. 

The  oil  groove  must  be  omitted  in  this  design,  but  this  matters  little 
if  oil  shields  are  used. 

Figure  399  is  the  design  of  a  disc  crank  used  by  the  Bass  Foundry 
and  Machine  Co.,  Fort  Wayne,  Ind.  The  disc  portion  is  considered  by 
some  to  have  a  more  finished  appearance,  the  rim  usually  being  polished. 

A  simple  arm  crank  with  counterbalance  is  shown  in  Fig.  400.  It  is 
proportioned  for  the  20  by  48  in.  Corliss  engine  designed  through  the 
book.  The  crank  of  Fig.  396  is  for  the  same  engine  and  is  calculated 
for  cast  iron,  but  Fig.  400  is  for  steel. 

Figure  401  shows  a  gas  engine  crank  with  pin  cast  integral,  and  is 
therefore  of  steel  casting. 


CHAPTER  XXVIII 

SHAFTS 
Notation. 

D  =  diameter  of  cylinder  in  inches. 
Ds  =  diameter  of  cylinder  when  some  standard  pressure  is  used 

(see  Par.  63,  Chap.  XII  and  Par.  72,  Chap.  XIII). 
d0  =  diameter  of  pin  or  shaft  determined  for  stress  relations.     Used 

especially  for  shaft  fit  in  hub. 
dp  =  diameter  of  crank  pin. 
ds  =  diameter  of  shaft  at  any  part  considered. 
h  =  thickness  of  crank  arm,  measured  axially. 
6  =  width  of  crank  arm. 
r  =  radius  of  crank  circle  in  inches. 
10  —  moment  arm  of  crank  pin  in  inches. 
1A  =  moment  arm  of  crank  arm  in  inches. 
1M  =  moment  arm  of  main  journal  in  inches, 
IP  =  length  of  crank  pin  in  inches. 
ls  =  length  of  main  journal  in  inches. 
p  =  maximum  unbalanced  steam  or  gas  pressure  in  pounds  per 

square  inch. 
P  =  maximum  pressure  on  crank  pin  per  square  inch  of  projected 

area. 
PM  =  mean  pressure  per  square  inch  of  piston  area,  including  inertia, 

per  cycle. 

Px  —  maximum  total  unbalanced  steam  or  gas  pressure  in  pounds. 
PP  =  total  unbalanced  force,  including  inertia,   acting  in  line  of 

stroke 

PL  =  thrust  along  connecting  rod  in  pounds,  including  inertia  of  re- 
ciprocating parts  and  of  rod  itself. 
PT  —  maximum  turning  effort  in  pounds. 
PR  =  force  in  pounds  acting  along  crank,  toward  shaft  center. 
PB  =  maximum  belt  pull  in  pounds. 
R  =  reaction  on  bearings  in  pounds.     Subscripts  correspond  with 

those  of  force  Px,  PT,  etc. 

RM  =  mean  reaction  on  bearing  during  cycle,  in  pounds. 
W  =  weight  of  flywheel  in  pounds. 

586 


SHAFTS 


587 


MB  =  bending  moment. 
M T  =  twisting  moment. 
MR  =  equivalent  bending  moment. 
A  =  total  deflection. 
6  =  deflection  per  inch  of  length. 
/  =  moment  of  inertia. 
E  =  modulus  of  elasticity. 
S  =  stress  in  general,  direct  or  combined. 
SP  =  stress  in  crank  pin. 
SA  =  stress  in  crank  arm. 
SM  =  stress  in  main  journal. 
SE  =  elastic  limit. 
Ss  =  shearing  stress. 
SD  =  direct  stress  in  crank  arm. 

/  =  factor  of  safety. 

C  and  K  are  constants  in  wear  formulas  in  Chap.  XI. 
Other  notation  on  figures.     All  dimensions  are  in  inches,  and  forces 
in  pounds. 

192.  Shaft  Types. — Two  general  types  of  crank  shaft  are  in  use; 
the  side-crank,  shown  in  Fig.  402  and  the  center-crank,  shown  in  Fig.  403. 


FIG.  402. — Side-crank  shaft. 


FIG.  403.— Center-crank  shaft. 

Center-crank  shafts  are  also  constructed  with  no  outer  bearing  for  small 
engines,  the  wheel  being  overhung  and  close  to  the  bearing.  Sometimes 
two  wheels -are  employed,  one  containing  the  governor. 


588  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

In  this  country  the  use  of  the  center-crank  is  mostly  confined  to 
engines  of  small  size  and  to  multi-cylinder  engines,  being  a  necessity  for 
the  latter.  Large  engines,  both  steam  and  internal-combustion,  are 
nearly  always  provided  with  the  side-crank  shaft.  On  the  continent, 
both  large  and  small  internal-combustion  engines  are  built  with  the 
center-crank  shaft. 

The  advantage  claimed  for  the  center-crank  is  an  even  distribution  of 
pressure  on  two  bearings,  and  the  advantage  from  the  standpoint  of 
strength  of  a  beam  supported  at  both  ends,  over  a  cantilever. 

It  is  obvious  that  an  outer  bearing  is  necessary  with  a  side-crank  shaft. 
For  many  classes  of  service  it  is  also  n'ecessary  with  a  center-crank  shaft, 
especially  for  large  sizes  with  heavy  wheels,  in  which  case  there  are  three 
bearings  to  keep  in  alinement.  Failure  to  do  so  results  in  excessive 
strain,  so  that  engineers  in  this  country  are  inclined  to  use  the  side- 
crank  shaft  where  feasible,  and  although  heavier,  stresses  may  be  cal- 
culated with  greater  accuracy. 

In  all  cases  of  design,  the  point  in  each  bearing  from  which  to  measure 
lengths  for  bending  moment  calculations  is  indeterminate,  and  certain 
assumptions  have  to  be  made.  For  the  side-crank  shaft  with  two  bear- 
ings, calculations  are  comparatively  simple,  as  is  also  the  case  with  center- 
crank  shaft  with  no  outer  bearing.  For  three  bearings,  and  for  multi- 
throw  shafts,  the  calculations  become  much  involved  if  an  attempt  at 
rigid  analysis  is  made.  The  Theorem  of  Three  Moments  (Clapyron's 
formula)  is  sometimes  applied  to  this  case  and  referred  to  as  an  exact 
method,  but  the  laws  of  bending  of  center-crank  shafts  are  very  complex ; 
in  fact,  part  of  the  deflection  is  due  to  torsion  of  the  crank  arms.  Were 
these  complications  due  to  shaft  form  taken  into  account,  assumptions 
must  still  be  made  due  to  the  resistance  to  bending  offered  by  the  bear- 
ings, and  the  nature  and  exact  location  of  the  reactions.  It  is  therefore 
certain  that  any  such  method  is  far  from  exact,  and  in  all  probability 
gives  no  better  results  than  the  simple  methods  employed  with  discretion, 
and  some  consideration  of  successful  practice. 

For  side-crank  shafts  an  analysis  which  has  given  satisfactory  results 
in  practice  has  been  employed,  and  this  is  checked  for  various  other  factors 
which  have  a  limiting  influence  on  design. 

For  two-bearing  center-crank  shafts  a  simple  analysis  is  also  made  for 
crank  pin,  arm  and  shaft  journal. 

For  center-crank  shafts  with  outer  bearing  the  portion  of  the  shaft 
between  main  and  outer  bearings  is  designed  as  for  a  side-crank  engine, 
while  the  cranks  are  designed  as  for  a  two-bearing,  center-crank  shaft, 
ignoring  the  remainder  of  the  shaft.  The  bending  moment  due  to  wheel 


SHAFTS  589 

load  would  be  practically  zero  at  the  point  of  maximum  bending  due  to  pis-  * 
ton  thrust,  and  it  is  unlikely  that  two  maximum  values  are  found  simul- 
taneously at  the  same  point.  It  may  be  that  wheel  load,  etc.  will  require 
a  larger  diameter  of  the  main  journal  than  given  by  the  simple  center- 
crank  analysis;  should  this  be  greater  than  the  required  crank  pin  diameter, 
they  should  be  made  equal. 

With  multi-throw  crank  shafts,  the  crank  next  to  the  delivery  end  of  the 
shaft  is  designed  as  a  single-throw  shaft,  with  the  addition  of  any  turning 
effect  from  the  other  cylinders.  This  is  best  determined  from  a  combined 
turning-effort  diagram,  omitting  the  end  cylinder  diagram.  It  is  usually 
comparatively  small,  especially  if  of  the  4-cycle  type,  and  may  be  pro- 
vided for  by  multiplying  the  moments  found  by  the  simple  method  by  a 
factor  a  little  greater  than  unity — possibly  1.1. 

On  account  of  greater  uncertainty  respecting  strains  in  center-crank 
shafts,  lengths  for  determining  moments  are  measured  more  nearly  from 
the  centers  of  bearings;  at  the  crank  pin,  however,  the  load  due  to  con- 
necting rod  thrust  is  more  symmetrical  about  the  center  of  the  pin,  and 
may  be  assumed  as  a  uniform  load  for  a  portion  of  the  pin  near  the 
center. 

Material — Shafts  for  large  engines  are  often  open  hearth  steel  forg- 
ings,  the  elastic  limit  of  which  may  conservatively  be  taken  as  38,000, 
as  given  in  Table  73,  Chap.  XXI.  Higher  grades  of  steel  are  sometimes 
used,  and  properties  of  some  of  the  alloy  steels  are  tabulated  in  Chap. 
XXI.  Allowable  bearing  pressures  sometimes  limit  the  stress,  so  that 
high  elastic  limits  are  not  required  unless  desired  for  their  higher  factor 
of  safety.  In  some  designs  however,  especially  where  few  bearings  are 
used  for  a  multi-throw  shaft,  the  ratio  of  bearing  pressure  to  stress  is  low, 
so  that  higher  permissible  stresses  are  a  great  advantage,  enabling  the 
use  of  smaller  diameters. 

The  side-crank  shaft  will  be  first  treated,  both  for  steam  and  internal- 
combustion  engines,  after  which  the  center-crank  shaft  will  be  analyzed, 
largely  in  connection  with  4-cycle  internal-combustion  engines,  where  it 
is  most  used. 

As  for  crank  pins,  the  factor  of  safety  may  be  determined  by  the  aid  of 
Par.  166,  Chap.  XXI.  From  Table  82  of  Chap.  XXI,  the  factor  for  single- 
acting  engines  may  be  taken  for  usual  conditions  as  3.45;  for  double- 
acting  steam  engines  as  5  and  for  double-acting  internal-combustion 
engines  as  4.2.  These  factors  are  based  upon  maximum  steam  or  gas 
pressure  only,  and  apply  to  crank  positions  on  or  near  dead  center.  As 
explained  in  Par.  166,  Chap.  XXI,  the  stresses  are  practically  repeated  for 
crank  positions  giving  large  turning  efforts,  and  the  factor  may  be  taken 


590 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


as  3,  with  perhaps  a  factor  of  judgment  of  from  1.1  to  1.2.  In  order  to 
reduce  deflection  it  is  better  to  use  a  higher  factor  for  shafts  with  con- 
siderable distance  between  bearings,  so  in  such  cases  the  factor  will  be  as 
in  Table  82,  Chap.  XXI. 

193.  Side-crank  Shaft. — Except  when  very  heavy  wheels  and  long 
shafts  are  used,  the  diameter  of  a  side-crank  shaft  may  often  be  deter- 
mined by  combining  the  bending  and  twisting  moments  due  to  the  piston 

thrust.  From  Formula  (11)  and  Table 
.  53  of  Chap.  XVI  it  may  be  seen  that 
when  the  connecting  rod  is  but  four 
cranks  long,  the  pressure  along  the 
rod  is  but  little  more  than  3  per  cent, 
greater  than  the  piston  thrust;  if  fric- 
tion were  considered  the  difference 
would  be  still  less.  Then  for  this 
discussion  it  may  be  considered  suf- 
ficiently accurate  to  neglect  the  angu- 
larity of  the  connecting  rod. 

The  older  rules  assumed  the  mo- 
ment arm  10  of  Fig.  404  to  extend  to  the 
center  of  the  bearing.  For  long  lines  of 

shafting,  or  where  ball-and-socket  bearings  are  used,  this  method  is 
probably  correct;  but  when  ball  joints  are  used  in  engine  bearings,  they 
do  not  allow  freedom  in  the  line  of  thrust.  It  is  then  probable  that 
the  point  of  maximum  bending  is  near  the  crank  end  of  the  bearing. 
The  author  has  assumed  for  many  years  that  it  is  at  the  point  where 
the  shaft  enters  the  crank.  The  shaft  is  practically  always  reduced  to 
form  a  shoulder  here;  then  if  this  reduced  diameter  be  solved  for  at  this 
point,  and  the  diameter  in  the  bearing  increased  %  to  Y±  in.  to  form  a 
shoulder,  the  equivalent  moment  arm  for  the  larger  diameter  will  extend 
a  short  distance  along  the  bearing.  By  this  method  it  is  not  necessary 
to  assume  stress  greater  than  that  indicated  by  the  laws  of  fatigue,  which 
was  necessary  with  the  old  assumption. 

Steam  Engine  Shaft. — For  engines  cutting  off  steam  later  than  one- 
half  stroke,  which  is  now  usual  when  overloads  are  being  carried,  the 
maximum  bending  and  twisting  moments  occur  when  the  crank  is  normal 
to  the  line  of  stroke.  The  bending  moment  is  then: 


FIG.  404. 


and  the  twisting  moment: 


MB  =  PX10 


MT  »  Pxr. 


SHAFTS  591 

The  value  to  be  used  in  these  formulas  is  not  strictly  Px,  but  PL,  the 
force  acting  in  the  direction  of  the  connecting  rod.     As  this  differs  little 
from  PP,  the  thrust  in  line  of  stroke  (including  inertia),  and  it  is  desired 
to  base  the  factor  of  safety  upon  Px,  the  latter  may  be  used. 
From  (22),  Chap.  XXI,  these  may  be  combined  thus: 


MK  =  0.35M*  +  QM\MB*  +  MT*  (1) 

Equating  with  the  modulus  of  section  of  the  shaft  gives: 

_  3/32M; 
d°       V~^S~ 

For  the  standard  cylinder  diameter  Ds  carrying  standard  unbalanced 
unit  pressure  p,  the  length  and  diameter  of  the  crank  pin  was  taken  as 
0.321)5  in  Chap.  XXVII.  The  crank  hub  was  taken  as  0.375D6,  Then 
for  the  design  shown  in  Fig.  404: 


10  =  +  0.375D5  =  0.535Z>a  (3) 

Let  r  =  1.2D8,  and  as  p  =  125  (Chap.  XXVII),   and  /  =  5,   by  sub- 
stituting these  values  in  (1)  and  (2),  d0  may  be  found  in  terms  of  Ds,  or: 

MR  =  98.6ZV 
and 

(^ 
(4) 


Taking  the  elastic  limit  SE  as  38,000,  S  =  7600  and: 

d0  =  QMDS 

Formulas  (4)  and  (5)  are  special  and  apply  for  the 
assumptions  made.  Their  use,  however,  will  give 
good  practical  results. 

Between  the  bearings  there  is  combined  bending 
and  twisting.  The  twisting  moment  is  the  same  as 
for  the  case  just  considered,  and  the  bending  may 
be  caused  by  weight  of  wheel  and  shaft,  weight  of 
gear  or  armature,  belt  pull,  or  magnetic  pull  of  arm- 
ature. Any  of  these  forces  may  be  referred  to  the 
bearing  by  taking  moments  about  the  other  bearing; 
then  a  resultant  may  be  found  graphically  as  in  Fig. 
405,  in  which  Rw  is  the  reaction  due  to  the  weight  and  RB  the  reaction 
due  to  the  belt  pull.  Total  reaction  R  may  include  all  forces  acting  be- 
tween the  bearings. 

The  central  part  of  the  shaft  is  usually  enlarged  to  take  the  wheel  or 
any  other  desired  mountings,  and  the  smaller  diameter  only  requires  cal- 


592 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


culation  for  strength  and  stiffness.  Referring  to  Fig.  406,  the  moment 
arm  10  may  be  taken  to  the  center  of  the  bearing  in  this  case,  as  there  is 
sometimes  freedom  of  movement  of  the  bearing  in  this  plane.  The 
bending  moment  is: 

MB  =  Rio 


From  (1)  the  equivalent  bending  moment  is: 
MR  = 


0.65  J(R10) 


+ 


(6) 


If,  after  adding  the  shoulder  to  d0  found  from  (2),  the  value  of  MR 
from  (6)  gives  a  greater  result  in  (2)  than  this,  it  must  be  used. 


FIG.  406. 

The  deflection  may  be  taken  as  for  a  cantilever  of  length  10. 

Rio*      S.SRlp*  =  5lo 
==  3EI  ~     Eds*      ''  12     V 


Then: 

(7) 


where  d  is  the  deflection  per  ft.  of  length.     Table  73,  Chap.  XXI  gives: 
E  =  29,000,000.    A  value  of  5  sometimes  given  is  0.003.    Then  from  (7)  : 

da  = 


All  three  of  these  methods  should  be  used  and  the  largest  diameter 
adopted.  By  using  a  lower  stress  in  the  first  method,  a  diameter  will  be 
obtained  which  may  be  ample  for  nearly  all  conditions  which  may  arise, 
but  it  gives  a  larger  shaft  than  is  required  in  a  good  many  cases,  increasing 
the  cost. 

A  further  check  should  be  made  by  the  formulas  of  Par.  52,  Chap.  XI, 
and  by  the  bearing  pressure  in  Table  19  of  the  same  chapter.  The  weight 
of  crank  and  part  of  connecting  rod  should  properly  be  included,  also  the 


SHAFTS 


593 


mean  piston  thrust.  The  resultant  mean  pressure  on  the  bearing  may  be 
found  for  each  stroke  of  the  cycle  and  the  average  of  these  taken  as  the 
value  of  P  in  Chap.  XI. 

It  is  not  claimed  that  these  values  give  limits  beyond  which  it  is  never 
permissible  to  go,  but  it  is  well  to  keep  within  or  near  these  figures  if 
possible. 

A  common  ratio  of  length  to  diameter  of  steam  engine  bearings  is  2. 
A  bearing  length  may  sometimes  be  made  twice  the  diameter  of  shaft  by 
the  first  method  even  though  the  actual  diameter  must  be  greater  accord- 
ing to  the  second  or  third  methods,  providing  the  check  on  bearing  pres- 
sure and  wear  by  Chap.  XI  is  satisfactory. 

In  some  cases  the  shaft  is  enlarged  immediately  outside  the  bearing. 
The  crank  fit  and  bearing  may  then  be  proportioned  for  the  piston  load, 
and  the  considerations  of  friction  and  wear  treated  in  Par.  52,  Chap.  XI. 


VilS  Vip 

Y//////V////A  Y///////////////A//////////////////////////A 


,4  ^ 

~  ~ 


FIG.  407. 

As  an  example  of  application,  assume  the  20  by  48  in.  Corliss  engine 
mentioned  in  Chap.  XII  and  elsewhere  through  the  book,  to  drive  a 
300  kw.  railway  generator.  The  data  for  this  generator  was  taken  from 
an  old  table,  but  will  answer  for  the  problem.  The  weight  of  the  arma- 
ture is  20,500  Ib.  The  wheel,  determined  by  the  method  of  Chap.  XVIII, 
weighs  33,000  Ib.  The  two  eccentric  hubs,  governor  pulleys,  wheel  hub, 
and  limiting  lines  of  generator  are  shown  in  Fig.  407 

As  /  =  6  between  bearings,  S  •=  6330. 

The  reaction  of  the  wheel  at  center  of  main  bearing  is: 


=  33,000  X  94.5  = 

lo  / 


Of  the  generator: 


Rg= 


20,500  X  61.5 

AO  I 


38 


594  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  weight  of  shaft  may  be  neglected  at  first.     The  bending  moment 
due  to  total  reaction  R  (=  31,950,  or  say  32,000  Ib.)  is: 
MB  =  32,000  X  31.5  =  1,010,000  Ib. 

The  maximum  turning  effort  is  39,250  (approximately).     The  twisting 
moment  is: 

MT  =  39,250  X  24  =  945,000  Ib. 
From  (1)  or  (6) : 

MR  =  1,380,000. 
This  is  more  conveniently  found  from  (24),  Chap.  XXI.     Now  from  (2) : 


/3 
~\ 


2  X  1,380,000 


TT  X  6330 

This  gives  some  idea  of  the  necessary  diameter,  but  we  know  it  will  be 
some  greater.  In  the  shaft  fit  (5)  gives: 

d  =  0.51  X  20  =  10.2  in. 
With  a  M-in.  shoulder  the  diameter  of  bearing  would  be  : 

d  =  10.5  in. 

With  this  diameter  as  a  basis  let  k,  the  ratio  of  length  to  diameter, 
be  2.  The  length  may  be  taken  as  21  in. 

Checking  for  deflection  by  (8),  with  reaction  R  gives: 

d  =  0.175v/32,000v/31.5  =  13.2  in. 

As  the  weight  of  the  shaft  was  not  included  it  is  better  to  take  the  diam- 
eter between  bearing  and  hub  fit  as  13.5  in.  It  is  desirable  when  taper 
keys  are  used  in  the  flywheel  hub,  to  increase  the  diameter  so  that  the  key 
will  clear  the  smaller  portion  of  the  shaft.  If  straight  keys  are  set  in 
the  shaft  this  is  not  necessary.  In  this  case  we  will  make  the  diameter 
through  wheel  and  generator  fit  16  in.,  an  increase  of  2^  m-  From  the 
generator  table,  the  range  of  shaft  diameter  is  from  13  to  16  in.,  so  this 
is  allowable. 

The  reaction  of  the  shaft  at  the  main  bearing  is  4000  Ib.  The  reaction 
of  the  crank  is  3700  and  of  the  connecting  rod  675  Ib.  The  total  reaction 
producing  bending  moment  is  due  to  wheel,  generator  and  shaft,  the  effect 
of  the  latter  being  safely  taken  as  a  concentrated  load  at  the  center;  this 
reaction  is  nearly: 

R  =  36,000  Ib. 

The  reaction  causing  bearing  friction  is  the  resultant  of  this  plus 
crank  and  rod  effect,  the  mean  load  due  to  piston  thrust  and  inertia.  The 
latter  at  maximum  cut-off  is  32,500,  and  as  this  acts  normal  to  the  other, 
the  mean  reaction  causing  wear  is : 

RM  =  V40,4002  +  32,5002  =  51,800  Ib. 


SHAFTS  595 

The  main  bearing  may  have  any  dimensions  from  10.5  by  21  in.  to  13.5 
by  27  in.,  as  far  as  strength  is  concerned.  By  trying  several  values  by  the 
formulas  of  Par.  52,  Chap.  XI,  we  will  take  12  by  24  in.  and  check.  This 
does  not  change  the  reactions  perceptibly,  but  lengthens  the  moment  arm 
10  to  33  in. 

For  strength: 

MB  =  36,000  X  33  =  1,190,000 
MT  =  39,250  X  24  =  945,000. 
From  (1)  or  (6) : 

MR  =  1,520,000. 

From  (2)  

,  =    3/32  X  1,520,000  =  13  45  in 
\      TT  X  6330 

which  is  safe,  as  13.5  in.  was  taken. 
For  deflection: 

From  (8) : 

ds  =  0.175v/36,000\/33  =  13.9  in. 

This  is  a  little  larger  than  the  dimension  taken,  but  the  moment  arm  is 
probably  less  than  33  in.,  and  there  is  some  variation  allowed  in  the 
deflection  factor  used,  which  was  a  mean  value. 
Bearing  pressure. 

=    51,800 


12  X  24 
Wear. 
From  (4),  Chap.  XI: 

C  =  Q'262  X  5*f°0  *Jgg  =  56,300. 

From  (6),  Chap.  XI: 

v       0.512  X  51,800    /100 

-       -       : 


The  values  of  C  and  K  are  greater  than  the  tabular  values,  but  these 
factors  are  more  or  less  arbitrary  and  may  be  increased  under  good  con- 
ditions. In  determining  P,  C  and  K,  the  maximum  load  pressure  was 
assumed  in  the  cylinder. 

The  shaft  is  drawn  to  scale  in  Fig.  407.  There  must  be  fillets  in  all 
places  where  the  shaft  changes  diameter.  A  smaller  diameter  might 
have  been  used  if  stronger  material  had  been  assumed,  but  the  modulus 
of  elasticity  is  practically  the  same  for  all  different  steels,  and  this  is  what 
influences  the  deflection. 

The  eccentrics  and  straps,  governor  pulley,  etc.  were  neglected.     The 


596  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

shaft,  crank  and  connecting  rod  are  often  omitted;  if  bearing  pressure 
and  the  constants  for  wear  are  based  upon  such  omissions  it  matters  little. 

The  10  in.  of  length  between  the  outer  bearing  and  the  swell  could  no 
doubt  be  reduced,  equalizing  the  load  more  nearly  between  the  two 
bearings.  It  is  possible  that  a  more  detailed  study  would  show  the  same 
for  the  21  in.  at  the  crank  end,  shortening  the  shaft  slightly. 

Internal-combustion  Engine  Shafts. — The  maximum  bending  and  tor- 
sional  strains  do  not  occur  at  the  same  time  in  these  shafts,  and  d0 
should  be  computed  for  maximum  bending  stress  at  dead  center,  and  for 
combined  bending  and  twisting  at  the  position  of  maximum  turning 
effort. 

At  the  dead  center  (1)  gives: 

MB  =  MB. 
Then  this  may  be  used  in  (2) .  At  the  position  of  greatest  turning  effort : 

MB  =  PL10 
and 

MT  =  PTr 

where  PT  is  the  turning  effort  and  PL  the  thrust  along  the  rod.  PL  will 
be  used  for  this  part  of  the  discussion,  although  the  numerical  value  of 
PP  may  actually  be  used  with  small  error.  The  position  of  PL  and  the 
value  of  PT  may  be  found  from  the  crank-effort  diagram.  Giildner  finds 
this  position  40  degrees  from  the  head-end  dead  center  for  his  reference 
diagram,  with  a  value  of  PL  =  0.7P*,  and  PT  =  0.5P*. 

In  Fig.  197,  Chap.  XVI,  the  maximum  turning  effort  is  at  45  degrees. 
PL  =  0.53P*  and  PT  =  0.43P*.  PX  is  the  total  maximum  gas  pressure, 
while  PP  is  the  total  pressure  at  any  point  including  the  effect  of  inertia. 
The  value  of  PL  and  PT  taken  from  Chap.  XVI  include  the  inertia,  while 
Giildner 's  values  do  not;  they  would  apply  at  starting  and  are  safer. 

Combining  the  twisting  and  bending  moments  in  (1),  d0  may  be 
found  from  (2).  The  analysis  for  stress,  deflection  and  wear  between 
the  bearings  are  the  same  as  for  the  steam  engine  shaft. 

Assuming  standard  pressure,  proportions  of  crank  pin  and  length  of 
shaft  fit,  a  special  formula  may  be  derived  as  was  done  for  the  steam 
engine.  In  deriving  a  special  formula  for  the  internal-combustion  engine 
side-crank  in  Chap.  XXVII,  it  was  assumed  that  the  length  of  crank  fit 
a  was  0.5D5;  then  taking  k  as  unity,  (12)  of  Chap.  XXVII  was  derived; 
or,  letting  dP  be  the  pin  diameter  (and  also  the  length  for  this  case)  : 

dP  =  1P  =  QA73DS. 
Then: 

10  =  0.737D.S 


SHAFTS  597 

Further  assume  that  P  =  400,  /  =  4.2  and  S  =  9000.     At  dead  center: 

Px  =  ^D2  X  400 

and 

TT  X  400  X  0.737 


This  is  the  only  moment  acting  in  this  position.     Taking  Gtildner's 
values  at  the  position  of  maximum  turning  effort: 


7T 


PL  =  0.7  X    D/  X  400  =  220D 


s 
<± 

PT  =  0.5  X  jlV  X  400  =  157IV. 

Then: 

MB  =  162ZV 
and 

MT  =  126ZV 
From  (1) : 

MR  =  182ZV. 

This  is  less  than  MB  for  the  dead-center  position,  so  the  latter  must  be 
used;  then  from  (2): 

d0  =  0.64As  (9) 

As  this  is  much  greater  than  the  value  given  by  (5)  for  the  steam 
engine  there  is  less  likelihood  of  the  computations  between  bearings  giving 
a  greater  diameter  than  with  the  steam  engine,  although  the  wheel  of 
the  latter  may  not  be  as  heavy.  As  mentioned  in  connection  with  steam 
engine  shafts,  the  diameter  may  be  increased  outside  the  bearing  if  a 
larger  diameter  is  required  for  the  wheel  load,  etc. 

Comparing  with  the  problem  of  the  steam  engine,  a  20-in.  double- 
acting  gas  engine  might  have  a  32-in.  stroke.  With  the  same  piston  speed 
it  would  run  150  r.p.m.  and  develop  approximately  one-half  the  power. 
A  tandem  engine  would  develop  the  same  power  as  the  steam  engine, 
and  aside  from  the  increased  inertia  effects  would  produce  no  greater 
strains  than  the  single-cylinder  gas  engine.  The  wheel  would  probably 
be  no  heavier  as  it  runs  at  higher  speed.  From  (9),  the  diameter  of  the 
shaft  fit  is: 

do  =  0.64  X  20  =  12.8;  say  12%  or  13  in. 

In  the  bearing  the  diameter  could  be  13H  m-  Otherwise  it  is  probable 
that  about  the  same  dimensions  could  be  used.  The  generator  from  the 
tables  used  before  weighs  but  17,000  Ib.  and  the  greatest  shaft  fit  is  14  in. 


598  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

It  is  likely  that  the  shaft  might  be  turned  to  this  diameter  at  the  genera- 
tor end  and  still  have  sufficient  stiffness. 

194.  Center-crank  Shafts. — For  either  steam  or  gas  engines  the 
center-crank  shaft  can  have  no  advantage  in  a  problem  like  that  of  the 
preceding  paragraph.  If  a  center-crank  is  specified,  the  pin  diameter 
may  be  determined  in  a  manner  which  will  be  given  later.  The  arms  may 
be  checked  for  bending  on  dead  center,  and  for  tension  and  bending 
when  normal  to  the  line  of  stroke.  In  this  latter  position,  due  to  the 
uncertain  effect  of  the  extra  bearing,  it  is  safe  to  ignore  it,  as  it  can  only 
cause  torsion  of  the  pin  by  undue  distortion  and  bearing  friction. 

In  multi-cylinder  marine  steam  engines  taking  power  from  one  end  of 
the  shaft — say  at  engine  No.  1.  The  bending  moment  on  the  pin  of 
No.  1  caused  by  the  maximum  turning  effort  of  this  engine,  may  be  com- 
bined with  the  twisting  moment  of  all  the  other  engines^  For  the  end 
journal  of  No.  1,  the  bending  moment  caused  by  the  thrust  of  this  engine 
must  be  combined  with  the  maximum  turning  moment  of  all  engines. 
This  can  be  determined  accurately  only  by  combined  turning-effort 
diagrams  at  maximum  load. 

The  formulas  developed  for  the  multi-throw  crank  for  internal- 
combustion  engines  may  be  applied  generally  to  steam  engines,  the  chief 
difference  being  in  the  ratio  of  twisting  to  bending,  and  the  position  of  the 
crank  when  principal  calculations  for  the  different  portions  are  made. 

Internal-combustion  Engines. — Calculations  show  that  the  crank  pin  is 
subject  to  maximum  stress  intensity  when  on  dead  center,  and  that  forces 
other  than  the  direct  piston  thrust  affect  it  but  little.  In  this  position 
the  moment  arm  of  the  main  journal  is  short,  so  that  it  receives  its  maxi- 
mum stress  at  or  near  the  position  of  maximum  twisting  moment,  the 
stress  being  combined  bending  and  torsional  stress. 

The  maximum  stress  in  the  arms  is  at  dead  center  position  usually, 
but  they  may  be  checked  for  combined  bending  and  twisting  in  the  posi- 
tion of  maximum  turning  moment. 

The  following  analysis  will  cover  these  points  in  as  simple  a  manner  as 
possible,  and  are  considered  as  reliable  as  a  more  complicated  method. 

In  treating  the  subject,  a  number  of  general  formulas  are  given  for 
the  various  straining  actions;  these  may  be  used  as  a  check  upon  design 
after  the  proportions  are  assumed.  The  more  important  formulas  are 
marke(J  *,  and  from  these,  design  formulas  will  be  derived  in  terms  of 
factors  which  may  be  assumed,  or  taken  from  practice.  These  are 
usually  sufficient  for  design  if  proper  values  are  assigned. 

Provision  is  made  in  the  more  general  formulas  for  an  overhung  wheel 
and  pull  of  belt.  Their  resultant  PG  may  be  resolved  into  any  required 


SHAFTS 


599 


components,  the  sign  of  which  may  be  easily  determined.     Should  there 
be  an  outer  bearing,  the  effect  of  PG  may  be  taken  as  one-half. 

To  get  a  better  idea  of  the  forces  and  reactions  dealt  with  in  the 
general  formulas,  the  skeleton  drawing  of  Fig.  408  is  given.  The  reac- 
tions are  all  shown  acting  at  the  same  point  in  each  bearing,  although 


i 


FIG.  408 


this  need  not  always  be  so  assumed;  the  point  is  often  indeterminate 
and  will  be  given  in  what  follows  without  reference  to  the  length  of  the 
bearing. 

The  twisting  moment  due  to  other  cylinders  is  also  given  in  the 


FIG.  409. 


general  formulas,  although  for  most  multi-cylinder  internal-combustion 
engines  this  is  not  great.  It  may  be  studied  from  the  combined  crank- 
effort  diagrams  of  Par.  176,  Chap.  XVI. 

In  order  to  give  further  notation,  Fig.  409  is  drawn.     The  thrust  of 
the  connecting  rod  PL  is  assumed  in  Fig.  409  as  a  load  distributed  over 


600  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

the  fraction  q  of  the  central  portion  of  the  crank  pin,  in  which  q  may  be 
any  desired  value  from  zero  to  unity.  Both  Figs.  408  and  409  contain 
notation  and  must  be  consulted  in  the  following  formulas. 

To  make  it  more  convenient  to  use  PG  in  certain  equations,  it  may  be 
referred  to  bearing  No.  1  as  an  equivalent  reaction  RE  which  acts  in  an 
opposite  direction  from  the  actual  reaction  RGi.  This  force  will  give  the 
same  moment  at  a  given  section  actually  given  by  P0.  From  Fig.  408  : 

(10) 


The  distances  A  and  B  may  be  taken  to  the  center  of  the  bearing  or  to 
some  assumed  point  given  by  10  of  Fig.  409. 

At  any  section  between  RE  and  the  connecting  rod  load  : 

MB  =  RM(B  -  lo)  =  P°A(-B-^ 

n 

Then: 

PGA(B  -  10) 


This  may  now  be  combined  with  RT,  RRi  or  RLi,  and  1M  or  1A  may  be 
used  instead  of  10  where  moments  are  desired  at  points  from  which  these 
measurements  are  taken. 

The  added  subscript  N  refers  to  components  of  RG  or  RE  normal  to  the 
crank,  and  P  refers  to  their  component  parallel  to  the  crank.  The  crank 
considered  is  either  for  a  single  cylinder,  or  for  the  engine  nearest  to 
where  the  load  is  applied,  and  will  be  referred  to  as  No.  1. 

The  force  PL  acting  along  the  rod  will  be  used  in  the  equations,  but  as 
previously  shown  PP  may  be  taken  as  its  numerical  value  without  serious 
error. 

For  PL,  PT  and  PR  see  Par.  99,  Chap.  XVI,  the  notation  for  these  quan- 
tities being  the  same. 

Crank  at  Dead  Center—  PL  =  Px  and  RL  =  Rx.  Also  f  =  3.45  for 
single-acting  engines  and  4.2  for  double-acting  engines. 

Crank  pin.     From  Px  : 

M*  =  R*ilo-?£'&  (12)* 

From  P0: 

MB  =  RE10  (13) 

The  resultant  of  these  may  be  found  graphically  and  used  for  the  bending 
moment.  For  vertical  engines  with  cylinder  over  crank,  (12)  and  (13) 
are  of  opposite  sign  and  in  line.  From  PG: 

MT  =  R02Nr  (14) 


SHAFTS  601 

From  the  turning  effort  of  all  cranks  when  No.  1  is  on  dead  center: 

MT  =  2PTr  *   (15) 

The  sum  of  (14)  and  (15)  may  be  combined  with  the  resultant  of  (12)  and 
(13)  by  means  of  (1);  then: 

MR=1^  (16)* 

Crank  arm.     From  Px: 

MB  =  RX11A  (17)* 

S  = 


From  the  turning  effort  of  all  cranks  when  No.  1  is  on  dead  center  : 

MB  =  2PT(r-^)  (19) 

o       6Mfi  /orkx 

8  =  ~w  (20) 

The  moment  given  by  (19)  is  always  normal  to  that  given  by  (17). 
From  PG: 

MB  =  RGZN(r-^)  (21) 

8  ~  ^  (22} 

'    hb*  (22) 

Also  from  P0: 


MB  =  REP1A  (23) 


From  turning  effort  of  all  cylinders  when  No.  1  is  on  dead  center : 

MT  =  2PT1A  (24a) 

From  PG: 

MT  =  RG^A  (25) 

For  the  algebraic  sum  of  (24a)  and  (25),  assuming  that  h  is  less  than  b, 
the  stress  at  center  of  the  shorter  side  from  Par.  163,  Chap.  XXI  is: 

Ss  =  MT(Zb  +  1.8ft)  (26) 

At  the  center  of  the  long  side: 

c        MT(3b  +  1.8ft)  ,_ 

*  =  -fttjjr-  (27) 

For  the  direct  load  due  to  Px: 

SD  =  %£  (28)* 


602  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

This  is  not  of  great  importance,  but  is  combined  with  (17)  in  the  design 
formula,  so  is  marked. 

In  combining  arm  stresses  three  combinations  may  be  made: 

(1)  The  algebraic  sum  of  (18),  (20),  (22),  (24)  and  (28). 

(2)  Combine  the  algebraic  sum  of  (18)  and  (24)  with  (27),  by  the  use 
of  Formula  (20),  Chap.  XXI. 

(3)  Combine  the  algebraic  sum  of  (20)  and  (22)  with  (26),  by  (20), 
Chap.  XXI.     Each  of  these  values  must  come  within  the  allowable  stress. 

Main  journal,  at  juncture  with  arm. 
FromP*: 

MB  =  RxllM  (29) 

From  P0: 

MB  =  RE1M  (30) 

From  turning  effort  of  all  cranks  when  No.  1  is  on  dead  center: 

MT  =  SPrr  (31) 


The  resultant  of  _(29)  and  (30)  may  be  combined  with  (31)  by  means  of 
(1);  then  S  or  ds  may  be  found  from  (16). 

Crank  in  Position  of  Maximum  Turning  Effort.  —  Factor  of  safety 
may  be  3.3  for  both  single-  and  double-acting  engines  except  when  wheel 
is  overhung,  then  4.2  should  be  used.  For  the  main  journal,  on  account 
of  uncertainty  of  the  point  of  application  of  load,  a  factor  of  judgment 
of  about  1.5  may  be  used. 

Crank  pin.     From  PL: 

MB  =  RLll0-^'&  (32) 

From  P0\ 

MB  =  RE10  (33) 

From  turning  effort  of  all  cranks  but  No.  1,  when  No.  1  is  at  its  position 
of  maximum  turning  effort: 

MT  =  SPrr  (33a) 


From  component  of  P0  normal  to  crank  : 

MT  =  RG2Nr  (34) 

The  resultant  of  (32)  and  (33)  may  be  combined  with  the  algebraic  sum 
of  (33a)  and  (34)  by  (1);  then  S  or  dP  may  be  found  from  (16). 

Crank  arm. 
From  PR,  the  component  of  PL  : 

MB  =  RR11A  (35) 


SHAFTS 


603 


For  the  effect  of  turning  effort,  No.  1  should  be  omitted.  It  will  be  in 
.the  position  of  its  maximum  effort,  however.  With  this  difference,  (18) 
and  (27)  apply  to  the  arms  in  this  position.  Formula  (28)  applies  by  sub- 
stituting RRi  for  RXI.  Then  by  taking  (35)  instead  of  (17),  the  analysis 
for  crank  on  dead  center  applies  to  this  case  for  the  arms. 

Main  journal  at  juncture  with  arm. 
From  PL: 

MB  =  RL11M  (36)* 

From  P0: 

MB  =  RE1M  (37) 

From  the  turning  effort  of  all  cylinders  when  crank  No.  1  is  in  posi- 
tion of  maximum  turning  effort: 

MT  =  SPrr  (38)* 


FIG.  410. 

The  resultant  of  (36)  and  (37)  may  be  combined  with  (38)  by  means  of 
(1) ;  then  S  or  ds  may  be  found  from  (16). 

Some  of  the  moments  counteract  the  others,  especially  due  to  PG. 
This  will  depend  upon  the  direction  of  belt  pull,  etc.,  and  if  this  is  to  be 
taken  into  account  in  designing  stock  engines,  the  worst  condition  should 
be  taken.  Bearing  pressures  and  wear  may  be  checked  by  Par.  52,  Chap. 
XL 

The  analysis  is  much  the  same  for  the  arrangement  shown  in  Fig.  410. 
The  reaction  Ri  will  be  greater  for  crank  No.  1;  therefore  the  bending 
moments  will  be  greater.  There  might  be  advantage  in  using  higher 
grade  material  for  a  crank  of  this  design  to  keep  dimensions  smaller. 

If  the  positions  of  the  bearings  may  be  located  with  reference  to  cylin- 
der No.  1,  reactions  RXI  and  RXz  may  be  found  and  calculations  made  as 


604  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

for  a  symmetrical  crank,  special  formulas  for  which  will  now  be  derived. 
Then  RXI,  instead  of  one-half  of  Px  would  be  a  larger  fraction.  As  this. 
relation  would  not  likely  be  standard  for  engines  of  different  size, 
special  formulas  would  perhaps  not  be  worth  while  in  this  case. 

Special  Design  Formulas  for  Symmetrical  Crank.  —  Referring  to  Fig. 
409,  let  IP  =  kdP,  h  =  xdP,  b  =  ydP  and  1M  =  mdP.     Then: 


7  71  /  I       X\   J 

h  =  IM  +  2  =  (™  +  g  j  d 
and 


Also 

P*=^  ;  (40) 

Crank  Pin.  —  Using  only  (12)  for  crank  at  dead  center: 


<«> 

Substituting  in  (12)  gives: 

(43) 


We  may  solve  for  dP  from  (16)  and  (43),  or  for  S  if  dimensions  are  known; 
but  it  is  more  convenient  to  consider  first  the  relation  of  S  to  P,  as  a 
compromise  must  sometimes  be  made  between  them.  Then  taking 
SP  from  (16)  and  P  from  (42)  : 

-  ,...* 

(44)* 

By  assuming  k,  x,  m,  and  <?,  the  ratio  SP/P  may  be  found  and  the  maxi- 
mum limit  of  one  fixes  the  value  of  the  other.  Then  dP  may  be  found 
most  conveniently  from  (42)  ;  or, 


Crank  Arm. — Combining  S  from  (18)  and  (28)  with  other  substitu- 
tions gives: 

irD*p 

&  A      ^~    

Substituting  the  value  of  dP  from  (45)  gives: 


SHAFTS  605 

All  factors  but  y  have  been  determined  already,  and  this  may  now  be 
determined  from  (47) ,  or  SA  checked  for  a  given  value  of  y.  P  should  be 
retained  as  already  determined,  but  it  may  be  that  the  value  of  S  from 
(44)  was  smaller  than  it  need  be,  in  order  to  keep  P  within  limit;  then  a 
higher  value  of  S  might  be  used  in  (47)  if  found  desirable. 

Main  journal  at  juncture  of  arm.  Maximum  stress  is  taken  at  posi- 
tion of  maximum  turning  effort  of  crank  No.  1,  for  which  (36)  and  (38) 
apply. 

Then:  RLl  =  -£ 


PT  may  be  taken  from  No.  1  crank  only.     The  bending  moment  is: 

p 
MB  =  -£  mdP 

and  the  twisting  moment  is: 

MT  =  PTr. 

They  may  be  combined  from  (1),  and  ds  found  from  (2).     By  taking 
Guldner's  values  for  his  standard  reference  diagram : 

PP  =  0.7P* 
and 

PT  =  0.5P* 

Also  assume  r  =  O.&D.     An  approximate  formula  may  now  be  derived 
by  substitution  as  before.     Without  going  through  the  derivation : 


/I      .     1.66/CP]  ,4oV 

V  +~ 


^£p[l       1.88^1 

The  value  of  P  is  for  the  crank  pin.  The  value  of  da  in  practice  ranges 
from  0.8  to  l.Odp,  usually  being  the  latter  in  automobile  engines.  One 
well-known  maker  of  multi-cylinder  gas  engines  makes  ds  and  dP  the  same 
on  their  4-cylirider  engines,  and  da  about  O.SdP  on  their  2-cylinder  engines. 

195.  Application  of  Formulas. — Formulas  (44)  to  (48)  may  be  used 
for  design — at  least  for  preliminary  calculations.  Should  there  be  a 
heavy  overhung  wheel  it  is  best  to  check  for  its  effect;  but  space  will  not 
be  taken  to  check  through  an  example,  and  it  probably  is  not  usually 
necessary.  When  extreme  lightness  is  not  desired,  a  factor  of  judgment 
may  be  employed  which  will  cover  minor  straining  actions. 

Giildner  says  that  the  factor  x  should  be  from  0.6  to  0.7.  It  is  some- 
times outside  these  limits.  The  factor  k  ranges  from  0.7  to  1.4,  the  more 
common  range  being  from  1.0  to  1.2.  The  factor  q  is  usually  ignored, 
the  load  being  considered  as  concentrated  at  the  center  of  the  pin.  It  is 


606  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

probably  more  nearly  the  truth  to  consider  it  as  a  uniform  load  over  a 
portion  of  the  pin  next  the  center,  say  one-half  of  the  length. 

The  factor  m  is  usually  such  that  the  moment  arms  1M,  1A,  and  10 
extend  to  the  center  of  the  bearing,  but  this  is  probably  not  a  correct 
measure  of  the  bending  moment.  It  is  probably  nearer  the  crank  arm, 
but  if  considered  too  short  in  computation  for  diameter,  the  shaft  may 
not  have  the  required  stiffness.  For  preliminary  work  where  center 
distances  are  not  known,  m  may  be  taken  as  0.5;  should  the  finally 
determined  distance  to  the  center  of  the  bearing  be  greater  than  this, 
it  does  not  seem  likely  that  the  pin  would  be  made  weaker  by  lengthening 
the  bearing. 

The  factor  y  must  be  greater  than  unity,  and  should  be  relatively 
large  as  x  is  made  small.  A  minimum  value  of  1.2  may  be  assumed. 

By  assuming  these  factors,  special  formulas  may  be  derived  for 
standard  engines  of  cylinder  diameter  DS}  or  the  factors  may  be  measured 
from  actual  engines  from  which  stresses  and  bearing  pressures  may  be 
readily  determined;  examples  of  both  kinds  will  be  given. 

Gas  Engine.  —  Assume  a  single-acting  gas  engine  with  a  maximum 
pressure  of  400  Ib.  per  sq.  in.  gage.  Let  x  =  0.6,  k  =  1.2,  m  =  0.5 
and  q  =  0.5.  Also  let  /  =  3.45,  which,  if  SE  =  38,000,  gives  S  =  11,000. 
Then  (44)  gives  for  the  pin  : 

Sp  -  Q  c 
Iv- 


or, P=  =  11601b. 

y.o 

From  (45)  : 

dP  =  0.475As. 

For  the  arm,  assume  y  =  1.2;  then  from  (47)  : 

SA  =  7550 
and 

/  =5. 

For  the  main  journal  the  maximum  stresses  are  more  uncertain  and  it 
is  well  to  use  a  factor  of  judgment;  this  will  be  made  1.5,  giving  /  =  5 
and  S  =  7600.  Then  from  (48)  : 

ds  =  QA3DS 
or 

^  =  0.91. 
dp 

The  value  of  10  is  0.81DS. 

By  taking  p  =  360,  S  =  14,000  and  10  =  0.9DS,  Guldner  obtains: 

dP  =  0.45Z)S. 


SHAFTS  607 

Had  he  used  p  =  400,  this  would  have  been  QA75DS  as  just  found  with  a 
stress  of  11,000  Ib.  Gtildner  assumes  the  load  concentrated  at  the  center 
of  the  pin  and  the  moment  arm  10  measured  from  the  center  of  the  bearing. 
It  is  no  doubt  safer  to  make  these  assumptions,  but  probably  not  so  near 
the  truth. 

In  the  problem  just  considered,  a  higher  elastic  limit  would  permit  a 
higher  bearing  pressure  on  the  pin  and  still  be  within  the  limit.  This 
might  also  be  accomplished  by  decreasing  the  value  of  k. 

Diesel  Engine. — By  scaling  the  drawings  of  a  certain  Diesel  engine  the 
following  values  were  found:  k  =  1,  x  =  0.5,  y  =  1.35,  m  =  0.65,  10  = 
l.Sldp  and  dP  =  0.606DS.  The  value  of  q  will  be  taken  as  zero.  Then 
assuming  p  as  500,  (42)  gives: 

P  =  1070  Ib. 
Also  (44)  gives: 

SP  =  8.4  X  1070  =  9000  Ib. 

Taking  SE  as  38,000,  /  =  4.22;  this  gives  a  factor  of  judgment  over  3.45 
of  1.22. 

From  (47) : 

SA  =  8570  Ib. 
and 

/  =  4.43 

This  gives  a  factor  of  judgment  of  1.28. 
From  (48),  if  SM  =  SP: 

ds  =  0.564D.S  =  0.93dP. 
These  diameters  were  made  the  same,  which  gives: 

SM  =  7300 
or, 

/  =  5.2. 

This  gives  a  factor  of  judgment  of  1.5. 

These  values  may  be  taken  as  fairly  representative.  The  measure- 
ments from  scale  drawings  from  three  makes  of  Diesel  engines  give  values 
of  dp  from  0.57  to  0.6061^. 

Automobile  Engine. — Values  from  a  well-known  automobile  engine 
are:  k •=  1.2,  x  =  0.427  and  dP  =  0.49D,s.  Assume  m  =  0.5  and  q  = 
0.5.  Then  from  (42),  substituting  the  value  of  dP: 

P  =  2.73p. 
From  (44) : 

SP  =  8.4P  =  23p. 


608  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

From  (47) : 

SAy  =  15.5P  =  42.3p. 

As  y  is  not  known,  it  will  be  solved  for,  assuming  SA  =  11,000. 
Then: 

p 

y  ~  260* 
These  values  are  given  in  Table  94  for  different  pressures. 

TABLE  94 


p 

P 

SP 

y 

300 

820 

6,900 

1.16 

350 

960 

8,050 

1.35 

400 

1,100 

9,200 

1.54 

450 

1,230 

10,350 

1.73 

If  y  is  less  than  the  tabular  values,  SA  is  greater  than  1 1 ,000. 

As  ds  is  equal  to  dP,  SM  is  less  than  SP  and  is  not  given.     Had  q  been 


Sect  ion  af" 


Section  B- 


Cra 
FIG.  411. — Mclntosh  and  Seymour  crank  shaft. 


Owernor  Wheel  Key 


Crank  Key 


fanerafvrHub  Key 


Fixed  (rov.  Ecc.  Key 


T         Eccentric  Link  Pin 


Crank  Shafr  Collar 


.Exciter  Pulley  Key 


FIG.  412. — Mclntosh  and  Seymour  shaft  details. 

neglected  and  10  taken  to  the  center  of  the  bearing,  the  stresses  would 
have  been  greater.  For  the  examples  given  it  may  be  seen  that  moderate 
bearing  pressures  and  stresses  obtain. 


SHAFTS 


609 


For  both  crank  pin  and  main  journal,  the  wear 
may  be  checked  by  the  formula: 


C  = 


PMND 
4.3 


P_ 

kp 


(49) 


This  was  derived  from  (1),  Chap.  XI,  the  allowable 
values  of  C  being  given  in  Table  19  of  that  chapter. 
196.  Designs  from  Practice. — Fig.  411  shows  the 


FIG.  414. — Crank  with  counterbalance. 

design  of  a  shaft  used  on  the  Mclntosh  and  Seymour 
Type  F  steam  engine.  The  outer  bearing  is  a  ring- 
oiled  bearing.  Grooves  are  turned  in  the  shaft  at 
each  end  of  the  outer  bearing  to  prevent  oil  from 
traveling  along  the  shaft  to  generator  or  exciter, 
the  latter  being  located  on  the  end  of  the  shaft 
extending  beyond  the  outer  bearing.  Details  of 
these  grooves,  with  other  details,  are  shown  in  Figs.  Ill 
and  412. 

The  center-crank  shafts  used  as  illustrations  thus 
far  have  been  drawn  to  scale.  A  6-cylinder  automobile 
engine  shaft  with  seven  bearings  is  shown  in  Fig.  413. 

-if-5 


A  crank  shaft  with  counter  balance  forged  integral 
— taken  from  Giildner — is  shown  in  Fig.  414. 

Collars. — In  Fig.  407  a  collar  was  shown  on  the  end 
of  the  shaft  beyond  the  outer  bearing.  A  standard 
collar  used  by  the  author  on  Corliss  engine  shafts 
is  shown  in  Fig.  415;  dimensions  are  given  in  Table 
95,  taken  from  the  following  formulas: 


=  4  +  2  in.     c  =  1.25ds  +  1.5  in. 


FIG.    413.— Franklin 
crank  shaft. 


39 


610  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  95 


<* 

b 

c 

^ 

6 

3 

9 

% 

7 

3 

10K 

% 

8 

3 

UK 

% 

9 

3 

13^ 

% 

10 

3M 

14 

1 

11 

•    .      3^ 

15^ 

1 

12 

3K 

16^i 

1 

13 

3K 

17K 

1 

14 

4 

19 

IK 

15 

4 

20H 

iSf 

16 

4 

21  M 

IK 

17 

4 

22K 

IK 

18 

*K 

24 

IK 

CHAPTER  XXIX 

FRAMES 
Notation. 

PP  =  total  pressure  in  line  of  stroke,  including  inertia. 
PN  =  total  pressure  on  guide,  normal  to  line  of  stroke. 

P  =  force  in  general,  in  pounds. 

PX  =  total  maximum  pressure  on  piston  due  to  steam  or  gas  pressure 
only. 

p  =  maximum  unbalanced  pressure  per  square  inch  in  cylinder. 

S  =  stress  per  square  inch  in  general. 
SB  =  bending  stress. 
SD  =  direct  stress. 
M  =  bending  moment. 

A  —  area  of  section  in  square  inches. 

I  =  moment  of  inertia. 

c  =  distance  from  neutral  axis  to  extreme  fiber,  in  inches. 

D  =  diameter  of  cylinder  in  inches. 

D8  =  diameter  of  cylinder  when  some  standard  pressure  is  assumed 
(see  Par  63,  Chap.  XII). 

d  =  diameter  of  bolt;  in  some  cases  at  root  of  thread  if  this  is  the 
weakest  section. 

n  =  number  of  bolts  involved  in  a  discussion. 

H  =  coefficient  of  friction. 

197.  Stresses  in  Engine  Frames. — While  serving  the  same  general 
purpose,  the  requirements  of  engine  frames  differ  widely,  depending  upon 
the  type  of  engine  and  class  of  service.  In  large  heavy-duty  engines, 
there  must  be  mass  to  furnish  stability  and  absorb  vibration.  Cast 
iron  is  the  material  best  adapted  to  such  frames.  In  marine  engines, 
lightness  is  important,  and  such  frames  are  often  built  of  steel  bars. 

For  center-crank  engines  the  forces  act  symmetrically,  and  frame 
stresses  are  comparatively  easy  to  determine.  Let  Fig.  416  be  a  diagram 
of  a  vertical  engine.  This  is  taken  as  there  are  no  restraining  forces 
due  to  foundation,  acting  on  the  frame  proper.  The  force  PP  is  the 
direct  force  tending  to  sever  the  frame.  The  guide  pressure  P#  tends 

611 


612 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


to  push  the  frame  from  its  vertical  position;  this  acts  at  the  distance  lx 
from  any  point,  producing  a  bending  moment  PNlx  at  that  point.  This 
may  be  resisted  by  the  cylinder  bolts  if  a  trunk  piston  is  used,  by  the 
foundation  bolts,  by  the  bolts  holding  the  frame  together,  and  by  any 
section  of  the  frame.  The  sum  of  the  stress  produced  by  PP  and  PN 
is  the  total  stress,  and  its  maximum  value  depends  upon  the  cut-off  in  a 
steam  engine.  Inertia  and  gravity  of  course  have  their  effect  as  may  be 
seen  in  Chap.  XVI. 

Neglecting  the  stress  due  to  screwing  up,  the  stress  in  the  bolts  is 
usually  repeated,  as  they  are  in  tension  only.  But  it  usually  is  reversed 
in  frame  sections  as  may  be  seen  from  Par.  166,  Chap.  XXI. 


FIG.  417. 


An  application  was  given  in  connection  with  cylinder  bolts  in  Par.  171, 
Chap.  XXII.  Considering  the  bolts  which  hold  the  A-frame  to  the 
base  in  Fig.  416  the  tension  in  the  bolts  on  the  right  side  is: 


The  tension  due  to  PP  is : 


PP 
2' 


The  total  tension  in  the  bolts  at  the  right  side  is  therefore: 

P  =  J?^    ,   PP 
a  2 


(1) 


Next  assume  a  section  of  a  horizontal,  cast  iron  frame,  neglecting  the 
effect  of  the  foundation.  This  is  shown  in  Fig.  417.  The  bending 
moment  is  due  to  PN  and  is  PNlx  as  before.  The  section  is  symmetrical 


FRAMES  613 

about  axis  xx  but  not  about  axis  yy;  furthermore,  the  direct  load  PP  is 
applied  at  a  distance  y  from  the  neutral  axis  of  the  section.  This  pro- 
duces a  bending  moment  PPy.  The  total  bending  moment  on  the  section 
then  is : 

M  =  PNlx  +  PPy  (2) 

The  bending  stress  is: 

S     =  — 
The  direct  stress  is: 


Or,  the  total  stress  is: 

-/-+X  (3) 

From  Par.  166,  Chap.  XXI,  the  factor  of  safety  for  single-acting  engines, 
for  a  cast  iron  frame  is  6;  for  double-acting  steam  engines  it  is  12,  and  for 
double-acting  gas  engines  10.8.  A  factor  of  judgment  may  be  used  if 
desired.  Frames  symmetrical  about  one  axis  may  thus  be  checked  with 
a  reasonable  amount  of  satisfaction. 

Side-crank  Frames. — Assume  a  side-crank  engine  with  a  section  like 
one-half  of  Fig.  417.  There  is  now  added  a  bending  moment  PNz  to  be 
resisted  by  the  modulus  of  section  about  axis  zz.  The  maximum  stress 
would  probably  be  at  the  upper  right-hand  corner  of  the  section.  The 
bending  moment  about  axis  yy  is  given  by  (2)  and  may  be  denoted  by 
MY]  about  axis  zz  the  bending  moment  is : 

Mz  =  PPz  (4) 

The  total  stress  is  then : 

'    +T         •  0) 


With  the  form  of  section  sometimes  used,  and  with  the  load  applied 
eccentrically  about  two  axes,  it  is  questionable  whether  the  stress  rela- 
tions given  by  (5)  hold  with  great  accuracy. 

Besides  the  bending  and  direct  stresses  in  a  side-crank  frame,  there  is 
a  torsional  stress  produced  by  PN.  It  is  absolutely  useless  to  attempt 
even  an  approximation  to  finding  the  torsional  stress  in  such  a  section. 
An  ample  factor  of  judgment  must  be  used  to  allow  for  this,  possibly  as 
high  as  2. 

Still  further   considering  the  forces  acting  on  a  side-crank  frame, 


614 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Fig.  418   shows   a   diagram   in  plan.     The  notation  is  all  contained  in 
the  diagram  and  applies  only  to  it.     Equating  moments: 

Pa  =  P!& 
or, 


Pi  = 


Pa 


Equating  couples: 


PA  =  PA. 


tC1 


FIG.  418 


FIG.  419. 


These  may  be  considered  as  resultant  couples,  the  actual  forces  being 
reactions  at  the  foundation  bolts,  and  friction.  They  are  indeterminate 
if  the  frame  is  attached  to  the  foundation  at  more  than  two  points,  which 
is  always  the  case.  The  bearing  jaw  receives  the  force  P  +  PI,  but 
tension  in  the  frame  is  only  P  when  PI  is  applied  as  shown.  If  PI  were 
applied  near  the  cylinder,  the  tension  in  the  frame  would  be  P  +  PI. 

Figure  419  is  a  side  elevation  of  the  same  engine. 
Equating  couples: 

PC  = 
Or,  as  PC  =  PJ,*  also 


Force  P3  may  be  supplied  by  weight  of  engine  and  by  downward  pull  of 
the  foundation  bolts;  it  is  also  indeterminate  if  the  frame  is  held  to  the 
foundation  at  more  than  two  points.  If  a  belt  wheel  or 
gears  are  used,  other  couples  may  be  introduced,  increasing 
the  resultant  couple  P3Z3. 

Figure  420  is  an  end  elevation.     Equating  couples: 


The  couple  P^a  produces  combined  bending  and  twisting, 
the  amount  depending  upon  the  manner  of  holding  the  frame  to  the 
foundation,  and  the  rigidity  of  the  latter. 

In  Figs.  418  to  420,  P  and  P4  are  forces  acting  within  the  engine,  due  to 
the  resistance  of  the  engine  shaft  to  turning;  forces  PI,  P2  and  PS  are 


FRAMES 


615 


reactions  on  the  frame  by  the  foundation.     If  transmission  is  by  belt, 

ropes  or  gears,  other  reactions  occur  and  other  couples  are  formed. 

A  consideration  of  the  foregoing  will  show  how  impossible  it  is  to 


030' 
Z61 

18 
14 


ZOO 


300 


400 

s 

FIG.  421. 


500 


600 


analyze  stresses  in  frames  of  this  type  with  any  degree  of  accuracy.     For 

this  reason  but  little  analysis  is  usually  attempted,  a  common  method 

being  to  divide  the  maximum  total  pressure  by  a  low  stress  to  determine 

the  required  area.     A  factor  of  judgment  of  from  2.5  to  5  is  sometimes 

used  with  this  method,  the  larger  values  being  for  the  smaller  engines. 

The  chart  of  Fig.  421  was  constructed  partly  from  data  found  in  Kent's 

Mechanical  Engineers'  Pocket-book  and  credited 

to  F.  A.  Halsey.     The  values  of  S  are  not  strictly 

the  stress,  but  merely  a  factor  of  design.     It  may 

be  called  the  nominal  stress.     This  chart  was  used 

by  the  author  in  Corliss  engine  design  for  several 

years.     It  of  course  applies  to  the  smallest  section 

subject  to  the  complex  loading.     For  the  neck  of 

the  frame  carrying  the  flange  which  connects  with 

the  cylinder,  the  applied  force  is  more  direct  and  a 

higher  stress  may  be  used.     A  factor  of  judgment  of 

from  1.25  to  1.5  may  be  used  with  the  factors  obtained  from  Par.  166, 

Chap.  XXI. 


616 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Ribs  on  flanges  are  objectionable,  and  it  is  better  to  increase  the  thick- 
ness of  a  flange  and  leave  it  plain.  Figure  422  gives  a  flange  and  neck  de- 
sign which  has  been  satisfactory  on  Corliss  engines.  The  following 
empirical  formulas  were  used,  B  being  the  diameter  of  the  stud  and  D  the 
diameter  of  the  cylinder.  The  notation  applies  only  to  Fig.  422. 


and 


F  =  0.075D  +  0.75 
T  =  0.045D  +  0.85 


(6) 
(7) 


Formula  (7)  gives  stresses  nearly  as  low  as  Fig.  421  for  a  steam  pressure  of 
125  Ib.  per  sq.  in.;  but  much  higher  pressures  may  be  used,  the  stress  in- 
creasing proportionally. 

In  heavy-duty  frames,  the  mean  stress  in  a  section  of  the  frame  be- 
tween the  guides  and  main  bearing  may  be  made  much  less  than  given  by 
Fig.  421,  especially  for  the  style  shown  in  Figs.  426  and  430.  The  weak- 
est portion  is  the  top  wall  and  the  side  wall  next  the  engine  center  line, 
and  these  should  be  somewhat  thicker.  It  may  be  well  to  include  in 
the  area  assumed  to  carry  the  load,  only  the  area  inclosed  within  a  cer- 
tain circle  of  radius  r,  as  shown  in  Fig.  417,  the  choosing  of  this  radius 
to  be  based  upon  experience.  Then  the  necessary  area  would  be: 

'  •        •       A-£  .  '  .         (8) 

^  To  sum  up,  this  method 

is  not  very  scientific,  but  it  is 
probable  that  many  engines 
are  successfully  running  today 
with  frames  which  did  not 
receive  even  the  analysis 
given  here.  It  perhaps  is  a 
little  more  satisfactory  if  re- 
sults are  checked  by  the  com- 
bined direct  and  bending 
stress  given  by  (5).  In  this 
the  twisting  moment  is  neg- 
lected and  a  factor  of  judgment  must  be  allowed;  if  this  is  as  high  as 
2,  the  check  may  be  assumed  satisfactory. 

Locomotive  frames  are  made  of  steel  castings  in  this  country,  and  the 
simple  formula  (8)  is  used  in  determining  main  dimensions.  As  already 
stated,  S  cannot  be  considered  as  the  actual  stress,  nor  is  it  as  applied 
to  locomotive  frames.  •  A  portion  of  a  frame  is  shown  in  Fig.  423. 


FIG.  423. 


FRAMES 


617 


Through  AA, 
Through  BB, 
Through  CC, 


A  = 


A  = 


to 


2500  2700 
3000  t0  3200* 
4300  t0  4500* 


FIG.  424. 


198.  Main  Bearings. — Gen- 
eral bearing  dimensions  are  dis- 
cussed in  Chap.  XI  and  Chap. 
XXVIII.  There  is  compara- 
tively little  which  permits  cal- 
culation in  the  main  bearing  of 
an  engine,  but  a  simple  discus- 
sion will  be  given  of  a  few  items. 

The  general  construction  may  be  seen  from  some  illustrations  given  in 
the  following  paragraph. 

Bearing  Jaw. — The  jaw  may  be  considered  as  a  cantilever  with  a 
maximum  load  Px  acting  at  the  center  of  the  shaft,  and  a  moment  arm 
I  as  shown  in  Fig.  424;  the  arm  I  must  be  taken  so  as  to  give  a  maxi- 
mum stress.  The  modulus  of  section  is  7/c,  c  being  taken  to  the  inside, 
or  tension  side  of  the  jaw.  Then: 


or, 


(9) 


A  strong  cap  is  provided  which  no  doubt  adds  greatly  to  the  strength  of 
the  jaw,  but  it  is  well  to  neglect  it  in  the  check. 

Wedge  Adjusting  Bolts. — The  hook  bolt  is  sometimes  used,  and  a 
simple  analysis  will  be  given.  A  wedge  and 
bolt  are  shown  in  Fig.  425.  No  engine  adjust- 
ments should  be  made  while  the  engine  is  run- 
ning, so  it  will  be  assumed  that  the  bolt  has 
only  to  hold  the  wedge  in  place. 

Let  k  be  the  taper  of  the  wedge  in  inches 
per  foot,  ju  the  coefficient  of  friction  and  n  the 
number  of  bolts  holding  the  wedge.  Also  let 
SB  be  the  bending  stress  and  SD  the  direct  stress.  Other  notation  is  on 
Fig.  425. 

The  maximum  total  load  on  the  bolt  is: 

kPx      2»PX      Px  ,  k 


L-, 


FIG.  425. 


Pi  = 


12n 


n 


(10) 


618  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

This  is  not  strictly  accurate  as  one  of  the  two  pairs  of  friction  surfaces 
is  slanting.     The  bending  moment  caused  by  PI  is: 

' 


Let  I  =  qd'}  then: 

The  direct  stress  on  the  bolt  is: 

S    =  — 
Then: 


Then: 


For  average  conditions  it  may  safely  be  assumed  that:  q  =  0.75,  k 

1.5  and  /*  =  0.036;  then  K  =  0.4725. 

Also 


_ 
4 

It  may  further  be  assumed  that  n  =  2.     Then  for  ordinary  use  : 

d  =  0.43Z)^|  (13) 

Assuming  a  standard  pressure  of  125  Ib.  as  has  been  done  in  other  parts  of 
the  book: 


If  S  =  7000,  d  =  0.059A,. 

If  S  =  10,000,  d  =  0.0493A*. 

For  internal-combustion  engines,  if  p  =  400  : 


This  would  give  excessive  sizes,  and  no  doubt  a  T-head  or  threaded  bolt 
would   be  used.     For  these  types,  q  =  0;  then  K  =  0.0675.     Then  if 

n  =  2: 

n 

(16) 


FRAMES  619 

Then  if  p  =  400  and  S  =  7000: 

d  =  0.039D5. 
If  S  =  10,000: 

d  =  0.0326D5. 

When  direct  load  only  is  applied,  d  is  the  diameter  at  root  of  thread. 

In  some  bearings  the  wedge  bolts  are  in  compression,  but  they  are 
usually  so  short  that  a  strut  formula  is  unnecessary. 

Cap  Bolts.  —  In  vertical  engines,  and  when  inclined  bearings  are  used, 
a  portion  of  the  piston  thrust  is  carried  by  the  bolts.  In  automobile  en- 
gines with  the  cap  on  the  bottom  of  the  bearing,  the  entire  piston  thrust 
is  taken  by  the  bolts  as  a  repeated  load.  In  the  common  form  of  horizon- 
tal engine  bearings,  no  load  is  theoretically  applied  to  the  cap  bolts.  The 
simplest  way  to  dispose  of  the  matter  is  to  assume  that  they  carry  the 
entire  piston  thrust.  This  gives  good  results  in  steam  engine  practice. 
If  the  bolts  are  unnecessarily  large  for  gas  engines,  a  factor  of  judgment 
less  than  unity  may  be  employed,  or  the  diameter  may  be  arbitrarily 
chosen  by  experienced  designers. 

Using  the  same  notation  as  for  the  wedge  bolts: 

ird2nS  _  irD2p 

~ir    ~T~' 

From  which: 


In  this  case  d  is  the  diameter  at  root  of  .thread. 

In  some  engines  n  =  2,  but  more  commonly  4  bolts  are  used. 
For  steam  engines,  if  p  =  125,  n  =  2  and  S  =  7000: 

d  =  0.095A,. 
If  n  =  2andS  =  10,000: 

d  =  0.079As. 
If  n  =  4and/S  =  7000: 

d  =  0.067^. 
If  n  =  4andS  =  10,000: 

d  =  0.05600. 

For  internal-combustion  engines,  if  p  =  400,  n  =  4  and  S  =  10,000  : 

d  =  Q.IDS. 

For  automobile  engines,  if  p  =  400,  n  =  4  and  SE  =  60,000  and  /  =  5, 
thenS  =  12,000  and: 

d  =  0.092D*. 


620 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


199.  Designs  from  Practice. — The  general  proportions  of  frames  may 
be  seen  from  builders'  catalogues,  and  books  which  are  largely  descriptive. 
A  few  examples  of  bearing  design  are  given  in  this  paragraph. 

Figure  426  is  the  bearing  built  by  the  Bass  Foundry  and  Machine  Co., 
Ft.  Wayne,  Ind.  There  is  a  double  wedge  adjustment  and  hook  bolts 


FIG.  426. — Bass-Corliss  main  bearing. 

are  used  for  wedges.  T-head  bolts,  set  in  pockets  are  used  for  cap  bolts, 
and  it  is  claimed  that  these  are  less  apt  to  break  than  studs  screwed 
into  the  frame.  The  top  box  is  not  babbitted  unless  desired  by  the 
purchaser. 

Figure  427  is  the  main  bearing  used  on  the  Lentz  poppet-valve  engine, 


FRAMES 


621 


built  by  the  Erie  City  Iron  Works.    It  is  a  chain-oiled  bearing  with  double 
wedge  adjustment. 

Figure  428  shows  a  bearing  used  on  the  horizontal  Diesel  engine  built 


FIG.  427. — Lentz  main  bearing. 


FIG.  428. — Allis-Chahners  Diesel  engine  bearing. 

by  the  Allis-Chalmers  Manufacturing  Co.,  Milwaukee,  Wis.  It  has 
double  wedge  adjustment,  the  wedges  being  forced  downward  by  a  special 
design  of  wedge  bolts. 


622 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


In  most  of  the  designs  shown,  the  bottom  box  may  be  rotated  about 
the  shaft  for  removal  without  disturbing  the  shaft  further  than  to  jack 


FIG.  429. — Girder  frame. 

it  up  slightly.  With  the  exception  of  Fig.  426,  there  is  no  provision  for 
changing  the  position  of  the  shaft  center,  the  wedges  being  only  to  take 
up  wear.  In  Fig.  426,  a  certain  amount  of  adjustment  of  the  shaft 
center  may  be  made  (the  line  showing  this  was  omitted  in  the  draw- 
ing). In  some  bearings  a  vertical  adjustment  is  also  provided. 


FIG.  430. — Heavy-duty  frame. 

Figures  429  and  430  are  Nordberg  Corliss  engine  frames  of  the  girder 
and  heavy-duty  types  respectively. 


CHAPTER  XXX 
FLYWHEELS 

200.  Introduction. — Chapter  XVIII  was  devoted  to  the  determina- 
tion of  the  weight  of  wheel  required  to  keep  speed  or  displacement 
within  fixed  limits.  In  this  chapter  the  various  straining  actions  will  be 
discussed  with  a  view  to  determining  proper  dimensions  to  make  the 
wheel  safe  against  rupture. 

Notation. 

P  =  force   in   pounds   applied   at   radius   R,   producing  bending 

moment  in  arms. 

PT  =  maximum  turning  effort  in  pounds. 
PF  =  sum  of  components  of  force  F2,  normal  to  hub  division. 
PR  =  resultant  of  forces  acting  on  shear  hub  bolts. 
w  —  weight  of  material  in  pounds  per  cubic  inch. 
WR  =  weight  of  rim  in  pounds. 
C  =  centrifugal  force  of  rim  in  pounds. 
T  =  tension  in  rim  in  pounds. 
FI  =  tension  in  one  arm  at  rim,  in  pounds. 
FZ  =  tension  in  one  arm  at  hub,  in  pounds. 
S  =  stress  in  general  in  pounds  per  square  inch. 
ST  =  stress  due  to  tension  in  rim. 
SPI  =  stress  in  arm  at  rim  due  to  centrifugal  force. 
SFZ  —  stress  in  arm  at  hub  due  to  centrifugal  force. 
SBR  =  bending  stress  in  rim. 
SBA  =  bending  stress  in  arm  at  hub 
SR  =  combined  stress  in  rim. 
SA  =  combined  stress  in  arm  at  hub. 
'     Si  =  tensile  stress  in  link,  or  in  hub  bolts ;  or  in  rim  bolts  at  root  of 

thread. 

Ss  =  shearing  stress  in  shear  hub  bolts. 
So  =  compressive  stress. 

TD  =  force  in  tons  per  inch  of  diameter  required  to  complete  forced 
fit. 


624  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TDL  =  same  as  TD,  but  per  inch  of  diameter  per  inch  of  length. 
R  =  radius  in  feet,  of  center  of  rim. 
Ro  =  radius  in  feet,  of  outside  of  rim. 
D  =  outside  diameter  of  wheel  in  feet  ( =  2R0) . 
TI  =  radius  in  feet,  of  inside  of  rim. 
r2  =  radius  in  feet,  of  outside  of  hub. 
L  =  length  of  arm  in  feet. 
a  =  length  of  crank  in  feet. 

hR    and  bR    =  depth  and  breadth  of  rim  section  in  inches,  respectively. 
hAi  and  bAi  =  depth  and  breadth  of  arm  at  rim,  in  inches. 
hAz  and  bAZ  =  depth  and  breadth  of  arm  at  hub,  in  inches. 
y  =  height  of  crown  in  inches. 
d  =  diameter  of  hub  in  inches. 

I  =  effective  length  of  hub  in  inches,  omitting  central  core. 
dT  =  diameter  of  rim  or  hub  bolts  at  root  of  thread. 
ds  =  diameter  of  hub  shear  bolts. 

Is  =  minimum  length  of  shear  bolt  in  inches,  taking  total  force. 
m  =  mean  depth  of  key  seat  in  inches. 
AR  =  area  of  rim  section  in  square  inches. 
AA  =  mean  area  of  arm  section  in  square  inches. 
AAi  =  area  of  arm  section  at  rim. 
AA2  =  area  of  arm  section  at  hub. 

AAS  =  area  of  arm  section  midway  between  rim  and  hub. 
AT  =  area  of  total  link  section  at  one  joint,  in  square  inches,  or  of  rim 

or  hub  bolts  at  root  of  thread. 

AC  =  total  link  area  at  one  joint  taking  compression. 
ZR  =  modulus  of  section  of  rim. 
zA1  —  modulus  of  section  of  arm  at  rim. 
zA2  =  modulus  of  section  of  arm  at  hub. 
/  =  moment  of  inertia  of  section. 
c  =  distance  of  extreme  fiber  from  neutral  axis. 
MA  =  bending  moment  on  one  arm. 
N  =  number  of  arms  in  wheel. 

Nv  =  virtual  number  of  arms  assumed  to  carry  entire  load. 
n  =  number  of  hub  bolts  or  rim  bolts  at  one  joint. 
V  =  velocity  in  feet  per  minute. 
t  =  time  in  seconds  required  to  bring  wheel  to  rest  with  a  constant 

force  P  applied  at  rim  center. 
kR  =  ratio  of  width  to  depth  of  rim. 
kA  =  ratio  of  width  to  depth  of  arm 
=  coefficient  of  friction. 


FLYWHEELS  625 

201.  Hoop  Stress.  —  It  is  shown  in  works  on  mechanics  that  unit 
radial  pressure  applied  to  an  indefinitely  thin  circular  ring  has  the  same 
effect  on  the  section  of  the  ring  as  the  same  unit  pressure  acting  normal 
to  a  diameter.  Then  if  the  total  centrifugal  force  of  such  a  ring  with  a 
radius  of  R  ft.  is  C,  the  unit  force  is  : 

C 


The  force  normal  to  a  diameter  tending  to  rupture  the  ring  is  then: 


7T 

The  hoop  tension  is  one-half  of  this,  or: 

~  2ir 
and  if  the  area  of  the  section  is  AR,  the  hoop  stress  is: 

rtl  /~1 

^  J.  U 


If  V  is  the  velocity  of  the  hoop  in  ft.  per  min.: 

WV2        irwA»Vz 
C  —  -         -  —          R  CX\ 

3600g«  "      150^ 
Then  from  (2) : 


Flywheels  are  usually  of  cast  iron,  for  which  w  =  0.26;  then: 

V2 

ST  =  37^000 

From  this  the  stress  may  be  determined  or  the  allowable  rim  velocity 
found. 

An  old  rule  for  the  maximum  velocity  of  cast  iron  flywheel  rims  is  "a 
mile  a  minute,"  and  this  is  pretty  generally  followed  today,  although 
6000  feet  per  min.  is  sometimes  allowed.  If  V  is  5280,  (5)  gives:  ST  = 
752  Ib.  This  is  a  low  stress,  but  in  case  of  overspeeding  from  any  cause, 
(5)  shows  that  the  stress  for  a  given  wheel  increases  as  the  square  of  the 
rotative  speed,  and  the  factor  of  safety  should  properly  be  based  upon  the 
speed.  The  velocity  V  is  usually  taken  at  the  outside  of  the  rim,  but  may 
be  taken  at  the  center  of  depth,  at  radius  R. 

The  foregoing  formulas  apply  only  to  very  thin  rings,  but  are  used  for 
flywheel  rims  of  the  usual  radial  thickness,  assuming  a  uniform  stress  dis- 
tributed over  the  section.  That  the  error  increases  greatly  as  the  ratio 
of  the  rim  thickness  to  wheel  diameter  increases  may  be  seen  in  the  follow- 
ing chapter,  but  it  may  be  safely  neglected  in  flywheel  design. 

40 


626  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  formulas  apply  also  to  unrestrained  rings  which  maintain  a  cir- 
cular form  at  all  speeds.  This  is  not  the  condition  of  the  flywheel,  the 
arms  modifying  the  stress  relations  to  a  considerable  extent.  Prof. 
Unwin,  in  his  Machine  Design,  gives  a  rather  elaborate  analysis  of  the 
stresses  produced  by  simple  rotation,  taking  account  of  the  influence  of 
the  arms,  upon  the  assumption  that  the  arms  extend  from  the  wheel 
center  to  the  center  of  the  rim  section. 

A  number  of  approximate  analyses  have  been  given  with  a  view  to 
simplification;  they  usually  depend  upon  some  constant  which  is  deter- 
mined by  guess,  or  by  comparison  with  Unwin's  analysis  for  a  given 
condition. 

Several  years  ago  the  author  substituted  more  readily  usable  terms  in 
Unwin's  formulas,  making  tables  of  the  more  unwieldy  quantities,  and 
taking  variable  angles  at  values  which  give  maximum  stresses.  Then  a 
factor  was  introduced  to  make  some  allowance  for  actual  arm  length 
upon  the  supposition  that  relatively  short  arms  stretch  less  and  produce 
higher  stresses.  With  these  changes,  the  Unwin  formulas  are  more 
easily  applied  than  most  of  the  approximate  methods,  with  the  probable 
advantage  of  greater  accuracy.  These  formulas  are  given  in  the  next 
paragraph.  No  allowance  for  bending  of  the  arms  is  included  in  this 
analysis,  but  this  will  be  treated  in  paragraphs  which  follow. 

202.  Unwin's  Formulas.  —  All  values  in  Unwin's  analysis  depend  upon 
the  arm  tension,  and  this  will  be  given  first;  as  determined  by  the  assump- 
tions made: 

_  wAAARV2  _  , 

2275(NAA  +  2irAR) 

This  is  Unwinds  value,  and  is  at  the  outer,  or  rim  end  of  the  arm.  In 
this  formula,  w  is  the  weight  per  cu.  in.,  AA  the  mean  arm  area  and  AR  the 
area  of  rim  section  —  both  in  sq.  in.  TV  is  the  number  of  arms,  equally 
spaced,  and  V  the  rim  velocity  in  ft.  per  min.  Let  subscript  1  refer  to 
the  under  side  of  the  rim  at  radius  ri,  and  2  to  the  outside  of  the  rim  at 
radius  r2;  let  3  refer  to  a  section  midway  between,  at  a  radius: 


2 
These  radii  are  all  in  feet.     Then  the  mean  area  of  the  arm  is,  very  closely  : 

/7x 
(7) 


AA2  +  AA3 


An  arbitrary  modification  may  now  be  made  by  multiplying  (6)  by  a 
factor: 


FLYWHEELS 


627 


where  R  is  the  radius  of  the  wheel  in  feet  (to  center  of  rim,  ac- 
cording to  Unwin),  and  L  is  the  arm  length  in  feet  (from  hub  to  rim). 
Then  where  the  arm  joins  the  rim: 


Fl  = 
The  stress  at  this  point  is: 


wAAARV2 


2275(NAA 


.11 


(8) 


(9) 


The  tension  at  the  hub  due  to  centrifugal  force  of  an  arm,  assuming 
the  center  of  gravity  at  the  center  of  the  arm  length,  is: 

wAALV2    ri  +  r2 

f A    = 


The  total  tension  at  the  hub  is: 

wAAV2 


F2=Fl+FA  = 
The  stress  at  the  hub  is : 


2275 


[ An IR    ,   L(ri  +  ra)1 

INAA  +  27rA«\L  "         S.5R 


A 


A  2 


The  maximum  hoop  tension  in  the  rim  is  at  the  arm,  and  is : 
The  hoop  stress  is : 


wARV*      F,        180 
T      "9660-    ~^CQi~N 


T 


(10) 

(11) 

(12) 
(13) 


180 

N 


The  values  of  cot  — ^-  are  given  in  Table  96  for  different  number  of  arms. 

TABLE  96 


N 

4 

6 

8 

10 

12 

cot^ 

A 

1.000 

1.732 

2.414 

3.078 

3.732 

The  maximum  bending  stress  in  the  rim  is  at  the  arm,  on  the  under 
side,  and  is: 


where  ZR  is  the  modulus  of  section  of  the  rim.     The  values  of  the  quan- 
tities in  brackets  are  given  in  Table  97. 


628  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  97 


N 

4 

6 

8 

10 

12 

N           180 

COt-^r 
7T                        N 

0.273 

0.  177 

0.132 

0.105 

0.087 

By  comparing  (4)  with  (12)  and  (13)  it  may  be  seen  that  the  maximum 
hoop  stress  is  reduced  by  the  arms.  However,  the  bending  stress  must 
be  added  to  obtain  the  maximum  rim  stress;  or, 

SR  =  ST  +  SBR  (15) 

In  like  manner  the  maximum  arm  stress  is  given  by: 

SA   =  SP2  +  SBA  (16) 

SBA  is  the  bending  stress  in  the  arm  at  the  hub  and  is  given  by  (27),  to  be 
derived  later. 

Should  it  be  desired  to  use  Unwin's  equations  as  originally  given, 
the  factor  \/R/L  may  be  omitted  in  (8)  and  (10). 

203.  Forces  Producing  Bending  Stresses  in  Arms.  —  The  maximum 
bending  moment  on  the  arms  will  either  be  due  to  the  maximum  turning 
effort  on  the  crank,  or  to  the  inertia  of  the  wheel  when  the  crank  is  on  dead 
center.  It  is  not  likely  that  these  will  work  together,  as  they  would  tend 
to  offset  each  other.  This  combined  effect  might  be  felt  in  the  teeth  of  a 
gear  on  the  shaft,  or  on  the  shaft  itself  if  the  wheel  is  between  the  crank 
and  the  gear. 

Let  PT  be  the  maximum  turning  effort  on  the  crank  in  Ib.  and  a  the 
crank  radius  in  feet.  The  maximum  force  applied  at  radius  R0  is: 


P  = 


Ro 


(17) 


If  a  heavy  rim  is  brought  to  rest  in  t  seconds  by  an  assumed  uniform 
force  P  applied  at  radius  R,  the  force  required  is  : 

WRV      WRV 


P  = 


1930Z 


(18) 


The  value  from  (17)  is  caused  by  belt  or  ropes  and  acts  in  a  direction  op- 
posite to  rotation;  while  the  force  in  (18)  is  overcoming  a  resistance  to 
gear  or  generator  and  acts  in  the  direction  of  rotation.  In  (17),  R0  is 
the  outside  radius  of  the  wheel.  In  (18),  V  may  be  taken  as  the  velocity 
at  radius  R. 

The  results  of  Formulas  (17)  and  (18)  depend  upon  the  value  assumed 
for  t,  and  the  number  of  arms  assumed  to  carry  the  load.  The  value  of  t 
may  be  taken  as  about  3,  although  this  is  not  based  upon  any  actual  tests. 
It  checks  fairly  well  with  heavy  wheels  used  for  rolling  mills  and  elec- 


FLYWHEELS 


629 


trical  machinery.     The  number  of  effective  arms  is  discussed  in  the  next 
paragraph. 

204.  Number  of  Arms  Carrying  Load. — There  have  been  various  ideas 
as  to  the  maximum  load  carried  by  one  arm,  or  virtually,  the  number  of 
arms  assumed  to  carry  the  entire  load.     It  is  certain  there  is  no  exact 
solution  to  the  problem.     The  nominal  stresses  allowed  by  the  Lewis  gear 
tooth  formula  are  high.     In  using  this  formula  for  large  cast  gears  some 
years  ago,  the  author  devised  a  formula  for  allowable  stress  in  which  two 
different  constants  were  used,  one  for  the  arms  being  25  per  cent,  greater 
than  that  for  the  teeth.     For  use  with  these  formulas  a  method  of  deter- 
mining the  number  of  arms  was  devised,  assuming  that  the  arm  opposite 
the  tooth  engaged  was  idle,  and  the  load  taken  by  the  other  arms  increas- 
ing in  direct  proportion  as  they  approached  the  teeth  which  were  in  mesh. 
Without  taking  space  for  the  derivation,  the  virtual  number  of  arms  tak- 
ing the  entire  load,  according  to  this  assumption  is: 

^-?  +  l  :.'.'.  (19) 

This  may  be  used  for  gears  when  the  allowable  stress  in  arms  and  teeth  is 
nearly  the  same  for  the  same  material.  Some  authorities  assume  the  rim 
so  stiff  that  the  load  is  distributed 
equally  among  the  arms.  The 
actual  condition  is  no  doubt 
somewhere  between  these  ex- 
tremes. 

Formula    (19)   was  modified 
for  flywheels  as  follows: 

(1)  For  belt  or  rope  wheels 
when  the  value  given  by  (17)  is 
greater  than  that  of  (18),  multi- 
ply (19)  by  2. 

(2)  For  belt  or  rope  wheels 
when  (17)  gives  a  smaller  value 
than    (18),    multiply    (19)    by 
2.5. 

(3)  For  flywheels  used  for  regulation  only,  Nv  =  N  —  1.     Then  use 
(18). 

These  rules  must  be  used  with  judgment,  as  it  is  not  claimed  that  they 
are  applicable  to  all  cases. 

205.  Bending  Moment  in  Arms. — Assume  an  arm  hinged  at  the  under 
side  of  the  rim  at  radius  n;  and  at  the  outside  of  the  hub  at  radius  r2  as 
shown  by  diagram  in  Fig.  431.     If  the  rim  revolves  while  the  hub  remains 


FIG.  431. 


630 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


stationary,  so  that  the  rim  end  of  the  arm  moves  through  a  very  small 
angle  5  from  its  original  position  relative  to  the  hub,  it  moves  through  the 
angle  a  from  its  original  position  relative  to  the  rim.  From  Fig.  431  : 


or, 


and 


a 


=  d 


4>  =  180  -  (a  +  0). 
From  the  properties  of  triangles: 

7*2  7*i  7*i 

sin  a  ~  siiT^  ~  sin  [180  —  (a  +  0)]  ~ 
For  the  very  small  angles  involved : 

a       sin  a       r<> 


sin  (a  +  0)  ~~  sin  5* 


sin  5 


(20) 


If  the  rim  and  hub  are  rigid,  the  bending  moments  at  the  two  ends  of 
the  arm  are  proportional  to  these  angles;  then  for  equal  bending  stress  at 
the  two  ends: 


where  ZA  is  the  modulus  of  section  of  the  arm.  If  the  arm  sections  at  the 
two  ends  are  similar  and  hA  is  the  depth  of  section  in  inches  in  the  di- 
rection of  motion : 


(22) 


Table  98  shows  this  relation. 


TABLE  98 


ri/r2  

1.500 

1.750 

2.000 

2.500 

3.000 

4.000 

5.000 

6.000 

1  145 

1  205 

1  260 

1  358 

1  442 

1  587 

1  710 

1  820 

A  rule  sometimes  used  for  wheel  arms  is  to  make  the  section  modulus 
at  the  hub  twice  that  at  the  rim,  and  this  usually  gives  a  well-appearing 
design.  With  these  proportions,  when  bending  only  is  considered  and 
when  TI/TZ  is  greater  than  2 — which  is  nearly  always  the  case — the  rim 
end  is  stronger  than  the  hub  end.  It  is  always  safe  to  take  zA2/zAi  as  1.5, 
the  minimum  value  of  Table  98,  and  this  gives  a  good  appearance. 

It  is  obvious  that  there  is  a  point  of  contra-flexure  at  which  the  bend- 
ing moment  is  zero,  and  that  this  moves  out  toward  the  rim  as  the  ratio 
a/ d,  and  therefore  r2/ri  is  made  smaller.  The  fact  that  the  rim  is  not 
rigid,  and  that  the  arm  is  more  flexible  toward  the  rim  due  to  the  reduced 


FLYWHEELS  631 

depth,  makes  it  apparent  that  the  point  of  zero  bending  moment  is  still 
nearer  the  rim  than  (20)  indicates.  It  is  therefore  obviously  safe  to  as- 
sume the  arm  as  a  cantilever  loaded  at  the  end,  which  may  be  considered 
as  at  the  inside  of  the  rim  at  radius  r\.  The  modulus  of  section  may  be 
taken  at  the  radius  r2,  which  is  at  the  outside  of  the  hub.  The  moment 
arm  may  be  taken  to  the  rim  center  for  (18)  and  to  the  outside  for  (17). 
The  bending  moment,  taking  P  from  (17)  or  (18)  is  then: 

MA  =  12(R  -  r2)  ~  (23) 

l\v 

The  maximum  value  of  P/NV  must  be  used.     For  a  heavy  rim  with  belt 
try  both  (17)  and  (18).     For  heavy-rimmed  balance  wheel  (18)  is  used. 
The  values  of  P  and  Nv  may  be  found  from  Pars.  203  and  204  respec- 
tively, and  R  in  (23)  taken  as  R}  R0  or  r\  as  desired. 
For  the  arm  at  the  hub: 

SBAzA2  =  MA  (24) 

From  this  either  SBA  or  zA2  may  be  determined.  Values  of  zA2  will  be 
given  in  Par.  207. 

206.  Working  Stresses. — A  flywheel  may  be  considered  as  subject  to 
repeated  loads  which  never  have  as  great  a  range  as  from  zero  to  full  load ; 
or  if  reversed,  the  percentage  of  maximum  load  is  small,  and  cast  iron  is 
much  stronger  in  compression  as  explained  in  Par.  159 ,  Chap .  XXI .  Safety 
decreases,  however,  as  speed  increases,  so  if  a  repeat  factor  is  taken  at  a 
comparatively  low  speed,  the  stress  at  this  speed  may  be  considered  the 
maximum  limit.  The  factor  may  then  be  increased,  or  the  allowable 
stress  decreased  gradually  up  to  the  maximum  allowable  speed. 

For  cast  iron,  which  is  most  commonly  used,  let  the  minimum  factor 
be  5;  if  the  ultimate  strength  is  taken  as  16,000  lb.,  this  makes  S  equal  to 
3200.    Let  the  velocity  at  stress  3200  be  1000  ft.  per  min.  and  assume  the 
stress  to  vary  as  the  cube  root  of  the  velocity  inversely. 
Then: 

32,000  /OCN 

,-,:••  *    "         s  =  -yr  '••  (25) 

This  is  an  entirely  empirical  formula  and  may  be  taken  as  the  maximum 
limit  of  allowable  stress  in  rim  or  arms  for  velocities  at  either  center  or 
outside  of  rim  from  1000  to  6000  ft.  per  min.  where  the  materia  is  cast 
iron.  Subtracting  from  this  the  hoop  stress  given  by  (13),  or  approxi- 
mately and  safely  by  (5),  leaves  the  allowable  bending  stress  in  the  rim 
which  limits  the  value  found  by  (14).  Subtracting  the  value  given  by  (11) 
from  that  of  (25)  gives  the  allowable  bending  stress  in  the  arm  near  the  hub. 
The  rim  section  is  determined  by  consideration  of  weight,  size  of  belt, 


632 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


or  the  stze  and  number  of  ropes.  In  determining  arm  dimensions  there 
are  various  empirical  formulas,  but  they  are  not  suitable  for  general 
application.  If  a  number  of  assumptions  are  made  in  (10),  a  simple 
formula  may  be  derived  which  will  assist  in  determining  arm  dimensions, 
and  if  desired,  these  may  be  more  carefully  checked.  Then  let  AA/AR  = 
0.25,  R/L  =  1.25,  L  =  0.8R  and  rl  +  r2  =  1.125fl.  If  zA2  =  2zAl  for 
similar  sections,  AA2  =  1.29  AA.  For  cast  iron,  w  =  0.26. 
Then: 


and 


To  facilitate  calculation  Table  99  is  given. 


TABLE  99 


(26) 
(27) 


V 

s 

SFZ 

N  =  6 

N  =  8 

AT  =10 

N  =  12 

4000 

2015 

448 

435 

415 

405 

4500 

1940 

570 

550 

525 

510 

5000 

1870 

700 

680 

650 

630 

5280 

1840 

785 

760 

725 

705 

5500 

1810 

850 

820 

785 

765 

6000 

1760 

1050 

980 

940 

910 

It  is  convenient  and  safe  to  take  V  at  the  outside  of  the  rim  in  finding 
S  and  SFZ.  It  is  probably  safer  not  to  design  wheels  for  a  velocity  much 
less  than  4000  ft.  per  min.  even  though  they  are  to  run  slower  than  this; 
this  is  especially  applicable  to  large  wheels. 

207.  Rim  and  Arm  Sections. — The  form  of  rim  section  is  determined 
by  the  service  required.  The  faces  of  belt  wheels  are  crowned  as  shown 
in  Fig.  437,  which  is  made  for  four  belts  of  different  widths.  There  is  no 
fixed  rule  for  the  height  of  the  crown,  although  formulas  are  sometimes 
given — more  usually  for  small  shop  pulleys.  A  formula  given  in  Leut- 
wiler's  Machine  Design  and  credited  to  Mr.  C.  G.  Barth  is: 


y  =  — 
32 


(28) 


where  bR  is  width  of  face  and  y  the  height  of  crown  in  inches. 

Formulas  for  the  ratio  of  wheel  face  to  belt  width  are  given  for  small 
pulleys,  but  are  generally  unsatisfactory  for  large  wheels.  From  %  to 
1H  in.  may  be  allowed,  depending  upon  the  size  of  the  wheel  and  condi- 
tion of  service. 


FLYWHEELS 


633 


There  are  a  number  of  designs  of  grooves  used  on  rope  wheels,  but 
they  are  usually  in  agreement  on  the  angle  of  groov3,  which  is  45  degrees. 
The  sides  against  which  the  rope  bears  are  usually  straight,  but  sometimes 
the  sides  are  curved;  it  is  claimed  for  the  latter  that  the  rope  is  more  apt  to 
turn  in  the  groove,  causing  more  even  wear  and  prolonging  its  life.  The 
multiple  system  is  mostly  used  for  main  drives  and  is  therefore  of  most  in- 


FIG.  432. — Engineers'  standard  groove. 


FIG.  433. — Cresson  standard  groove. 


terest  to  engine  designers.  Figure  432  shows  a  groove  known  as  the 
Engineers'  Standard,  designed  by  the  Jones  and  Laughlin  Co.  Table 
100  gives  dimensions  for  ropes  of  different  size . 


TABLE  100 


[8 


1 

IK 
IK 

2 


2% 

2^6 

3% 
3K 
4 
3K 

3^6 


K 


i* 

IK 


2% 


5/16 


IK 


2H 


i 

IK 


IK 

.2 


KG 

K 

KG 


1 

IK 
IK 


In  the  G.  V.  Cresson  Co.  catalogue  of  1907,  their  standard  for  the 
American,  or  continuous  system  has  a  groove  angle  of  45  degrees,  while 
for  the  English,  or  multiple  system  the  angle  is  30  degrees.  The  groove 
for  the  multiple  system  is  shown  in  Fig.  433.  The  ratio  of  the  tension 
on  the  tight  side  to  that  on  the  loose  side  is  greater  than  when  the 


634  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

angle  is  45  degrees;  the  power  transmitted  would  therefore  be  greater 
for  the  same  maximum  tension.  With  the  curved  groove,  an  old  rope, 
well  fitted  to  the  groove  would  settle  deeper  in  the  groove  where  the 
angle  is  greater,  while  a  new  rope  will  ride  the  groove  at  a  smaller  angle, 
thus  tending  to  divide  the  work.  The  author  used  a  groove  much  like 
the  Cresson  standard  for  a  number  of  years,  and  so  far  as  he  knows 
it  has  always  been  entirely  satisfactory,  and  especially  well  adapted  to 
heavy  work  and  large  overloads. 

As  the  rims  of  belt  and  rope  wheels  are  relatively  thin,  flanges,  and 
sometimes  ribs  are  used  to  increase  strength,  as  shown  in  Fig.  437. 
The  modulus  of  section  is  for  a  section  similar  to  a  T;  in  taking: 

,.-!        .:•;.--;:'  (29) 

c  must  be  taken  from  the  neutral  axis  to  the  edge  of  the  ribs — toward 
the  wheel  center.  The  section  should  be  taken  near  the  arm,  as  the 
maximum  bending  moment  is  at  this  point.  The  modulus  of  section  of 
a  rope  wheel  rim  may  be  found  by  the  graphical  method  of  Appendix  1. 
The  section  between  two  rope  centers  may  be  taken,  the  value  of  ZR  being 
for  the  sum  of  these  sections. 

Heavy  flywheels  usually  have  a  section  so  nearly  rectangular  that  the 
section  modulus  may  be  that  of  a  rectangle;  or, 


The  weight  of  the  rim  is  determined  by  one  of  the  methods  of  Chap. 

XVIII. 

If  this  is  WR: 

2ir  X  12RbRhRw  =  WR. 

If  6*  =  kRhR:  

~W  I     VF  ' 

h*  =  24wwbRR  =  \24irJLfl 

R  may  be  taken  to  give  a  certain  velocity;  then  the  outside  diameter  is: 

D  =  2R  +  hR  (31) 

If  D  is  to  have  a  certain  value,  R  may  be  assumed,  a  tentative  value 
of  hR  found,  and  adjustment  made  by  trial  and  error.  In  using  (30a), 
some  standard  value  of  kR  may  be  taken,  Halsey  preferring  %.  To 
check  the  weight  when  adjustments  are  made: 

WR  =  ir(l2D  -  hR)hRbRw  (32) 


FLYWHEELS  635 

For  cast  iron  wheels,  w  =  0.26,  and  (30a)  becomes: 


lfk  =  %: 

l'"';'  '""'"     (34) 

In  some  very  heavy  rims,  when  the  diameter  is  limited  by  a  gear 
shaft,  as  was  the  case  with  Fig.  436,  a  more  efficient  wheel  is  obtained  by 
making  bR  greater  than  hR,  so  long  as  the  latter  is  large  enough  to  prop- 
erly accommodate  the  links. 

In  determining  the  thickness  of  belt  wheel  rims,  the  first  form  of 
(30a)  may  be  used,  bR  having  been  determined  from  the  required  belt 
width;  R  may  be  taken  as  the  outside  radius.  The  value  of  hR  so  found 
would  give  less  than  the  required  weight,  and  crowning  reduces  this  still 
more  by  reducing  the  radius  to  the  center  of  gravity;  but  the  addition 
of  ribs  and  flanges  will  probably  compensate  for  this.  If  greater  accuracy 
is  desired,  the  weight  may  be  carefully  calculated  after  the  section  is 
designed.  In  finding  the  moment  of  inertia  of  the  section  it  is  best  to 
neglect  the  crown,  using  only  the  thickness  hR  found  from  (30a),  and 
the  ribs  and  flanges.  The  depth  of  rim  flanges  in  inches  may  be  made 
about  0.3D,  where  D  is  outside  diameter  in  feet.  The  thickness  of  the 
flange  may  be  about  0.2  times  the  depth  of  the  flange  if  this  is  not  greater 
than  the  rim  thickness. 

Arm  sections  are  usually  approximate  ellipses  drawn  by  circular  arcs. 
A  theoretically  exact  section  modulus  may  be  found  graphically  by  the 
method  of  Appendix  1.  It  is  usually  sufficiently  accurate  to  assume  an 
ellipse  and  this  simplifies  calculation.  It  will  be  assumed  in  this  discus- 
sion that: 

zA2  =  1.5zAl  (35) 

Also  let  the  sections  taken  along  the  arm  be  assumed  similar.     Then  if 
bA  =  kAhA; 

TrbA2h2A2       TrkAh3A2  irkAh3A1  ,     . 

Z^  =    "32"         ~32"    "  L5  "32~ 
From  this: 

hAi  =  ki»YDP  °'875^2  (37) 

Then: 


LA2 


=  4 


=   Q.7S5kAhA2  (38) 


636  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

And: 

Also : 


AAl  =  0.765AA2  (40) 

To  find  the  mean  area: 


and: 

AAS  = 

Then  from  (7),  the  mean  area  is: 

AA  =  0.707kAhA2*  (41) 

A  good  arm  proportion  is  when  kA  =  0.5;  then  the  following  special 
formulas  may  be  obtained: 
At  hub: 

zA2  =  0.0492/*3A2  (42) 

At  hub: 

AA2  =  0.392/^2  (43) 

At  rim: 

AA1  =  0.392^1  (44) 

The  mean  area  is  : 

AA  =  0.353/i2A2  (45) 

Equating  (42)  with  the  bending  moment  given  by  (23)  gives: 


h»  =  2.73  (46) 

\  &BA 

SBA  may  be  obtained  from  (27). 

208.  Hubs.—  The  analysis  of  hubs  given  in  Par.  3,  Chap.  XXVII 
may  be  used  for  wheel  hubs  and  will  not  be  repeated.  The  result  given 
by  (24)  of  that  chapter,  making  the  outside  hub  diameter  twice  the 
bore,  is  quite  common  practice.  With  this  ratio,  combining  (21)  and 
(22)  of  Chap.  XXVII,  gives: 

T        -    **S  (47} 

IDL  "1250 

where  TDL  is  tons  per  inch  of  diameter  per  inch  of  length  required  to  com- 
plete the  fit  if  the  hub  were  forced  on,  S  the  stress  and  n  the  coefficient 
of  friction.  This  formula  will  be  used  in  connection  with  the  discussion 
of  hub  bolts. 

If  I  is  the  effective  length  of  fit  on  the  shaft  and  TD  is  tons  per  inch 
of  diameter: 

TD=  TDLl  (48) 


FLYWHEELS  637 

To  simplify  equations,  especially  when  I  is  not  well  known,  TD  is  some- 
times used;  when  I  is  later  determined,  S  may  be  checked  by  (47). 
209.  Hub  Bolts.  —  Let  it  be  assumed  that  the  bolts  are  to  hold  the 
hub  to  the  shaft  with  a  pressure  equivalent  to  a  forced  fit  requiring  TD 
tons  per  inch  of  diameter  to  complete.  Then  if  d  is  the  diameter  of  the 
fit,  dT  the  diameter  of  the  bolt  at  root  of  thread,  Si  the  bolt  stress  and  n 
the  number  of  bolts  : 


TTfl 

Or: 


Taking  TD  =  5,  p  =  0.25,  n  =  4  and  Si  =  10,000: 

dT  =  0.637\/d  (50) 

The  bolts  may  be  selected  from  the  bolt  table  in  Appendix  2. 

The  resultant  of  the  force  F2  acts  on  the  hub  bolts  and  reduces  the 
value  of  TD]  this  resultant  is: 

PF  =  F22  sin  0  (51) 

where  0  is  the  angle  made  by  the  arm  with  the  diameter  at  right  angles 
with  the  bolts      Table  101  gives  values  of  S  sin  0. 

TABLE  101 

1  i 

X  6  8  10 


2  sin  e 

2.000 

2.704 

3.240 

Then  in  reality  the  equation  is: 

irdT2nSi  _ 

~~ 


The  value  of  TD  is  then: 

T  ^      nrdr'nffi       p  1 

TD  ~  20005  L~T~ 

Formula  (52)  may  be  used  as  a  check,  in  which  TD  must  be  a  positive 
quantity  capable  of  holding  the  wheel  firmly  to  the  shaft.  The  original 
value  of  TD  used  in  (49)  should  be  used  for  determining  stress  in  (47) 
and  (48),  as  the  effect  of  the  arms  does  not  reduce  the  hub  stress. 

Hub  bolts  are  usually  steel  bolts  having  a  nut  on  each  end  (commonly 
called  studs). 


638  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Shear  bolts  are  sometimes  used  in  large  wheels,  the  hubs  being  forced 
on  the  shaft.  Such  a  construction  is  shown  in  Figs.  436  and  437.  This 
style  of  fastening  cannot  be  designed  satisfactorily  without  drawing  to 
scale,  and  may  require  more  than  one  trial.  An  algebraic  solution  is 
clumsy  and  liable  to  error;  therefore  a  simi-graphic  method  is  easier  and 
better. 

Figure  434  shows  a  diagram  for  one  arm.  The  force  P/NV  at  the  rim 
causes  a  direct  force  equally  divided  among  the  three  bolts,  and  acting 
in  the  same  direction,  parallel  to  P.  The  force  F%  may  be  obtained  from 
(10)  and  the  resultant  of  these  two  forces  found  graphically  as  shown. 
Unless  x  is  much  greater  than  y,  the  bolt  for  which  the  diagram  is  drawn 
receives  the  maximum  load.  The  force  PY  acting  on  this  bolt  may  be 
found  as  follows: 


FIG.  434. 

Equating  moments  about  the  geometrical  center  of  the  bolts: 


Also: 

Px   =  Pv 

x         y 
or: 


..  ...,.„„ 

Substituting  gives: 

(53) 


This  may  be  combined  graphically  with  the  resultant  already  found, 


FLYWHEELS  639 

giving  the  final  resultant  PR.     This  is  resisted  by  the  bolt  in  double  shear. 
Then: 


_ 
Or: 


The  length  of  fit  in  either  hub  or  arm  must  be  : 


0.8  (54) 


This  construction  requires  a  driving  fit  of  turned  bolts  in  reamed  holes. 

As  with  the  arms,  the  maximum  assumed  stress  is  probably  seldom 
reached,  so  whether  the  stress  is  repeated  or  reversed,  it  is  probably 
through  a  small  range  and  of  an  intensity  much  less  than  the  maximum. 
A  factor  of  safety  of  3  may  be  assumed,  which,  if  the  elastic  limit  in 
shear  is  taken  as  29,000  as  in  Table  73,  Chap.  XXI,  gives  S8  =  10,000, 
nearly.  Sc  may  be  taken  as  12,000,  but  with  the  usual  construction  this 
will  not  be  reached.  Shear  bolts  are  also  usually  stud  bolts. 

210.  Rim  Bolts  and  Links.  —  These  must  hold  the  rim  tension  T 
given  by  (12),  or  more  simply  and  safely,  by  (1);  then: 

AT  =  ^  (56) 

Oi 

where  AT  is  the  total  area  of  the  link  sections,  or  the  area  of  bolts  at 
root  of  thread.     It  may  also  be  taken  as  the  bearing  area  of  the  link  head 
in  which  case  Si  =  Sc,  the  compressive  stress. 
Neglecting  the  effect  of  arms: 

A     -WA*V*  (57) 

9650S! 

As  the  stress  distribution  in  links  is  probably  not  uniform,  it  is  safer  to 
use  a  low  stress.     If  this  is  taken  as  5000  and  the  rim  is  cast  iron,  (57) 

becomes: 

A   172 

A     —          R  (W\ 

~  186,000,000 

The  link  may  be  proportioned  for  the  maximum  speed,  or  when  V  =  5280, 
and  the  dimensions  retained  for  all  speeds;  then: 

AT  =  0.154*  (58a) 

With  /  links,  the  bearing  pressure  may  be  twice  as  great  as  the  tensile 
stress  just  assumed;  the  area  for  this  is  then: 

Ac  =     -  =  0.0754*  (59) 


640  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

It  must  be  remembered  that  (58)  and  (59)  give  the  area  of  both  links  at 
any  section.     In  providing  for  ACt  allowance  must  be  made  for  clearance 
in  the  link  pocket,  as  the  projection  at  the  link  ends  will  not  bear  on  the 
surface  nearest  the  link  due  to  this  clearance. 
For  rim  bolts: 

irdT2n 


where  n  is  the  number  of  bolts  and  dT  the  diameter  at  root  of  thread. 
Then  (57)  becomes: 

fi     .  lwAR  far\\ 

UT  -  o7\/-o—  Iwi; 

or  \  Oin 

For  a  cast  iron  wheel  with  a  rim  velocity  of  5280  ft.  per  min.,  if 
>Si  =  7000: 


(61) 
\  n 

If  Si  =  10,000: 

^  (62) 

n 

Bolts  may  be  selected  from  Appendix  2.     As  with  the  hub  bolts,  stud 
bolts  are  generally  used,  having  a  nut  at  each  end. 

211.  Keys. — Standard  keys  are  often  used,  a  given  width  of  key 
always  being  used  with  a  given  shaft  diameter.  While  such  keys  are 
usually  ample  they  may  be  much  stronger  than  necessary  and  weaken 
the  shaft  unnecessarily.  It  is  always  well  to  check  a  key  in  engine  work. 
With  notation  already  used  in  this  chapter,  taking  m  as  the  mean  depth 
of  the  key  way  in  either  shaft  or  hub,  the  equation  of  moments  gives : 

Scml  -|  =  12P-^- 
From  which: 


The  hub  is  sometimes  cored  for  a  part  of  its  length,  through  which 
the  key  does  not  bear,  therefore  I  must  be  taken  as  the  actual  bearing 
length.  Sc  may  be  taken  as  about  10,000  Ib. 

212.  Methods  of  Construction. — Wheels  are  commonly  made  in  one 
of  the  following  ways: 

1.  Cast  in  One  Piece. — Small  wheels  are  made  in  this  way.  Some- 
times the  hub  is  split  to  relieve  shrinkage  strains;  after  boring  to  a  snug 
fit  it  is  clamped  to  the  shaft  by  the  hub  bolts. 


FLYWHEELS  641 

2.  Cast  in  One  Piece  and  Split  Apart. — This  method  is  sometimes  used 
with  small  and  medium-sized  wheels.     It  cannot  be  considered  very  good 
engineering.     Arms  sometimes  part  from  the  rim  when  the  wheel  is 
split,  which  leads  one  to  believe  that  large  internal  stress  may.  exist. 

3.  Cast  in  Halves  with  Planed  Joints. — This  is  a  good  construction  if 
properly  designed  and  cast,  and  is  used  for  large  wheels.    Figure  435  is  an 
example  of  this  design.     For  belt  and  rope  wheels  it  is  probably  better 
to  have  the  split  at  the  arms,  making  a  double  arm.     Such  wheels  are 
much  less  apt  to  be  broken  in  transportation,  and  are  no  doubt  stronger 
when  in  use. 

4.  With  One  or  Two  Arms  Cast  with  a  Segment  of  the  Rim. — This  type 
usually  has  the  arms  fastened  to  the  hub  with  shear  bolts.     A  design 
with  one  arm  cast  with  a  segment  is  shown  in  Fig.  436. 

5.  Built  up  from  Separate  Hubs,  Arms  and  Rim  Segments. — Such  a 
wheel  is  shown  in  Fig.  437.     It  is  likely  that  this  construction  gives  a  mini- 
mum of  internal  stress  when  properly  machined.     It  also  permits  ease  of 
shipment. 

A  series  of  oft-quoted  experiments  were  made  by  Prof.  Benjamin  on 
very  small  wheels  of  different  design  to  determine  their  relative  strengths, 
in  which  the  one-piece  wheel  proved  strongest.  It  does  not  seem  as  if 
this  applies  to  large  wheels,  and  it  is  very  doubtful  if  the  wheel  of  Fig.  437 
would  be  as  strong  cast  in  one  piece  as  it  is  at  present,  even  though  the 
feat  were  practicable  and  shipment  possible. 

The  relative  area  of  arm  and  rim  section  should  doubtless  be  in- 
fluenced by  the  mode  of  construction.  With  a  wheel  made  by  method  (4), 
and  even  more  by  (5),  the  arm  section  may  be  as  much  smaller  than  the 
rim  as  desired  if  computation  shows  it  strong  enough.  With  the  other 
methods  this  is  not  so,  due  to  internal  stress. 

While  cast  iron  is  the  most  common  material  for  wheels,  semi-steel  and 
sometimes  steel  castings  are  used.  If  the  latter  were  more  common,  report 
of  flywheel  " explosions"  would  be  more  uncommon.  Special  wheels  are 
built  up  of  steel  plates,  and  some  are  wound  with  steel  wire. 

A  wheel  was  built  by  the  Mesta  Machine  Co.  a  few  years  ago  for  a  rim 
velocity  of  10,000  ft.  per  min.  Neglecting  arm  influence,  the  hoop  stress 
from  (5)  would  be  2700  Ib.  The  wheel  was  cast  from  air-furnace  iron  with 
a  tensile  strength  of  30,000  Ib. 

213.  Application  of  Formulas. — The  weight  of  a  wheel  for  a  20  by  48 
in.  Corliss  engine  was  determined  in  Chap.  XVIII.  It  was  to  run  100 
r.p.m.  and  the  rim  weighed  20,500  Ib.  With  an  outside  diameter  of  16 
feet,  the  formulas  of  Par.  207  gives  a  section  15  in.  deep  and  10  in.  wide. 
The  latter  measurement  gives  a  weight  a  little  in  excess,  and  will  be 

41 


642  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

modified  slightly  in  the  finished  design  as  shown  in  Fig.  435.  The  sh#ft 
bore  of  16  in.  was  determined  in  Chap.  XXVIII,  and  the  outside  diam- 
eter of  the  hub  is  32  in.  The  length  of  hub  was  taken  as  18  in.  This 
might  perhaps  be  24  in.,  but  18  will  be  used.  The  number  of  arms  will 
be  taken  as  8. 

From  the  above  data,  as  many  additional  data  will  be  taken  as  possi- 
ble, and  are: 

R  =  7.37;    L  =  5.42;     rl  =  6.75;     r2  =  1.33;     V  =  4620;     t  =  3. 
From  (19),  Case  3:     Nv  =  7.     AR  =  150,  and  from  (30),  ZR  =  375. 

From  (18): 

20,500  X  4620  _ 

r  1930  X  3  1MUU  1D> 


M         12  X  6.04  X  16,400 

MA  =  — = —  =  170,000. 


32,000 

s  = 


From  (23) : 
From  (25) : 

From  (26)  : 

SF2  =  573  Ib. 
From  (27) : 

SBA  =  1247  Ib. 
From  (46) : 


3  170,000       t  A  . 
=  14  in. 


From  (42) : 

zA2  =  135. 
From  (43) : 

AAZ  =  76.5. 
From  (37) : 

hAi  =  12.25  in. 
From  (44) : 

AA1  =  58.7. 
From  (45): 

AA  =  69. 
From  (8)  : 

=  0-26  X  69  X  150  X  46202    17.37 

2275(552  +  940).        \5.42  ~ 
From  (9) : 

SFI  =        '7     =  350. 
Oo./ 


FLYWHEELS  643 

From  (10)  : 

0.26  X  69  X  46202r  150       5.42  X  8.08] 

~2275~      ~  LI492  +  8^T7^J  =  32>8C 

From  (11),  the  actual  value  of  SFz  is: 

e         32,800       ,OQ  „ 

S"2  =  ~76^~      428  lb" 
From  (12): 

86,000  -  24,900  =  61,100  lb. 


From  (13): 
From  (14): 


6  X  20,600  X  7.37  X  0.132 
-  ~~375~ 


From  (24),  the  actual  value  of  SBA  is: 

170,000 
=  —  135 

This  is  slightly  greater  than  the  assumed  value. 
From  (15),  the  total  rim  stress  is: 

SB  =  408  +  320  =  728  lb. 
From  (16),  the  total  arm  stress  is. 

SA  =  428  +  1260  =  1688  lb. 

This  is  about  87  per  cent,  of  the  allowed  stress  from  (25). 

Hub  Bolts. 
From  (50): 

dT  =  0.637\/16  =  2.55  in. 

The  nearest  safe  standard  bolt  diameter  from  the  bolt  table  of  Appendix  2 
is  3  in.     The  actual  diameter  at  root  of  thread  is  2.629  in. 
From  (51)  and  Table  101  : 

PF  =  32,800  X  2.704  =  89,000  lb. 
From  (52)  the  effective  value  of  TD  is: 

T  X  0.25  r^X  2.629*  X  4  X  10,000  __  1  _ 

TD  ~  2000  X  161  ~4~ 

This  is  tons  per  inch  of  diameter.     The  total  force  is: 
16  X  2.975  =  47.5  tons, 


644  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

which  far  exceeds  the  weight  of  the  wheel,  and  would  therefore  hold  it  if 
the  shaft  were  vertical. 
From  (48)  : 

2  975 
TDL  =    '        =  0.248  tons  per  in.  of  diameter  per  in.  of  length. 

\.2i 
From  (47),  the  hub  stress  due  to  this  force  is: 


It  must  be  assumed  that  the  total  hub  stress  is  4000  lb.,  the  value  used  in 
determining  hub  thickness,  the  difference  being  due  to  PF. 

Rim  Links.  —  Assuming  the  links  to  take  the  entire  load,  (58)  gives: 

150  X  462Q2  _ 
=  186,000,000    : 

The  area  of  one  link  section  is  8.6  sq.  in.  and  the  section  may  be  made 
2J4  by  3%  in.  As  the  bearing  pressure  is  twice  as  great,  the  head  may 
be  5%  in.  wide;  allowing  Y±  in.  on  each  side  for  clearance,  the  head  width 
may  be  6J^  in.  wide.  This  reduces  the  rim  section  by  the  area: 

2  X  6.75  X  2.25  =  30.3  sq.  in. 

The  original  rim  area  has  been  assumed  as  150  sq.  in.,  but  is  really  about 
135  sq.  in.  The  actual  hoop  stress  is  then  about  30  per  cent,  greater 
than  the  computed  hoop  stress,  but  the  bending  stress  is  affected  but 
little  as  the  metal  is  removed  from  the  center.  The  bending  moment 
midway  between  the  arms  is  one-half  as  great  as  at  the  arms.  The 
bending  stress  may  be  assumed  as  about  0.6$^,  or  190  lb.;  the  hoop 
stress  may  be  taken  as  30  per  cent,  greater  than  ST,  or  530  lb.,  the  sum 
being  720  lb.,  which  is  practically  the  same  as  SR  previously  found. 

If  there  were  two  rim  bolts,  and  they  were  assumed  to  carry  the  entire 
load,  their  diameter  would  be  3  in.  In  this  type  of  wheel  the  bolts  are 
used  for  convenience  of  construction,  so  they  may  be  made  smaller  than 
this,  perhaps  2  in.  in  diameter. 

Keys.  —  It  has  been  assumed  that  the  hub  is  cored  out  at  the  middle 
third  of  its  length,  giving  an  effective  length  of  12  in.  If  a  single  key  is 
used,  (63)  gives  for  the  mean  depth  of  key  way: 

12  X  16,400  X  16 
m  ='-  10,000X12X16  = 

This  may  be  made  \%  in.  and  the  width  of  the  key  3J^  in.  If  two  keys 
spaced  90  degrees  apart  are  used,  they  may  be  half  as  large,  or,  as  the 


FLYWHEELS 


645 


author  has  usually  done,  they  may  be  made  from  %  to  %  as  large — say 
\Y±  deep  by  2Ji  in.  wide. 

Total  Weight.— The  weight  of  the  wheel  is  approximately  32,675  Ib. 
Before  this  example  was  computed,  Table  62,  Chap.  XVIII  was  computed, 
giving  33,000  Ib.  as  the  weight;  this  is  1.6  times  the  rim  weight. 

Figure  435  is  a  scale  drawing  of  a  portion  of  the  wheel,  showing  con- 
struction. The  method  of  gradually  enlarging  the  arm  as  it  joins  the  rim 
has  been  used  by  the  author  on  many  heavy  wheels,  although  not  original 
with  him.  Holes  (not  shown)  are  cored  in  the  rim  are  for  jacking  the 


FIG.  435. 


wheel  around  for  the  purpose  of  valve  setting  or  to  get  the  crank  from 
dead  center.    They  are  spaced  about  6  in.  center  to  center. 

214.  Designs  from  Practice. — Figure  436  shows  a  wheel  built  by  the 
Bass  Foundry  &  Machine  Co.  It  is  18^  ft.  in  diameter  and  weighs 
140,000  Ib.  It  runs  75  r.p.m.,  giving  a  rim  speed  of  only  4350  ft.  per  min. 
It  was  built  for  a  geared  rolling  mill  drive  and  its  diameter  was  limited  by 
the  countershaft  carrying  the  gear.  To  increase  its  efficiency,  the  rim 
was  made  wide.  It  is  an  example  of  an  arm  and  rim  section  being  cast 
together,  the  arm  being  fastened  to  the  hub  by  shear  bolts.  Both  links 
and  bolts  are  used  at  the  rim,  the  former  being  designed  to  carry  the 
entire  load. 


646  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


—1 

_J 

E 

Q 

o  II  o  oj 

FIG.  436. — Bass-Corliss  rolling-mill  engine  wheel. 


FLYWHEELS 


647 


648  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Figure  437  shows  another  Bass  wheel.  It  is  a  double  wheel,  24  ft.  in 
diameter  with  a  10  ft.  face  crowned  for  four  belts.  It  is  a  good  example  of 
a  built-up  wheel,  each  arm  and  segment  being  cast  separately.  The  wheel 
weighs  about  200,000  Ib.  and  runs  80  r.p.m.,  giving  a  rim  speed  of  6000  ft. 
per  min.  The  wheel  is  driven  by  a  28  and  56  by  60  in.  cross-compound 
Corliss  engine.  The  bearings  are  18  by  36  in.  and  the  shaft,  made  of 
crucible  steel  by  the  fluid  compressed  process,  has  a  hole  8  in.  in  diam- 
eter through  its  entire  length. 

Reference 
Halsey's  handbook  for  machine  designers. 


CHAPTER  XXXI 

TURBINE  WHEELS 
Notation. 

d  =  diameter  of  wheel  bore  in  inches. 
Do  =  diameter  of  disc  in  inches  measured  to  center  of  gravity  of  rim 

and  blades. 

r0  =  radius  of  same  =  D0/2. 
r  =  radius  in  inches  at  any  point. 
t0  =  thickness  of  disc  at  radius  r0. 
t  =  thickness  of  disc  at  radius  r. 
WR  =  weight  of  rim  in  pounds. 
WB  =  weight  of  blades,  shrouding,  etc.  in  pounds. 
W  =  weight  of  rim,  blades,  etc.  carried  by  disc. 
P  =  load  per  inch  of  periphery  due  to  W. 
F  =  tangential  force  in  pounds  at  any  radius,  due  to  transmission 

of  power. 

E  =  modulus  of  elasticity. 

S  =  stress  in  general,  in  pounds  per  square  inch. 
So  =  assumed  tangential  stress  where  disc  and  rim  join. 
SB  =  equivalent  simple  tangential  stress  where  disc  and  rim  join. 
SL  =  limiting  tangential  stress  in  rim,  giving  same  strain  as  S0  in  disc. 
Sc  =  radial  stress  where  disc  joins  rim,  due  to  centrifugal  force  of  W. 
SD  =  shearing  stress  produced  by  F  at  any  radius  r  at  which  F  is 

calculated. 

SH  =  hoop  stress  in  thin  revolving  ring  of  diameter  D0. 
ST  =  total  tangential  stress  at  any  radius  r. 
SR  =  total  radial  stress  at  any  radius  r. 
fw  =  tangential  force  due  to  jet  velocity. 
fp  =  axial  force  due  to  jet  velocity.  T 

M  =  bending  moment  on  blades. 
Z  =  modulus  of  section  of  blades. 

n  =  number  of  blades  receiving  steam  at  one  time.     Also  exponent. 
I  =  moment  arm  of  blade  in  inches. 
m  =  the  reciprocal  of  Poisson's  ratio. 

649 


650 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


N  =  revolutions  per  minute. 
V  =  velocity  in  feet  per  second. 

a  =  area  in  square  inches. 
H  =  horsepower  developed  in  blading  of  wheel. 

q  =  fraction  of  the  rim  carried  by  itself. 

215.  In  all  impulse  turbines  the  blades  are  carried  upon  disc  wheels 
which  must  be  designed  so  that  the  maximum  stress  will  not  exceed 
certain  limits  at  the  intended  speed  of  rotation. 

The  hoop  stress  in  a  revolving  ring  is  given  by  Formula  (4),  Chap. 
XXX.  Taking  the  diameter  in  inches  at  the  center  of  gravity  of  rim 
and  blades,  the  formula  may  be  written  for  steel: 

(1) 


107    VIOO 

For  rim  velocities  ordinarily  used  for  flywheels,  the  value  of  SH  is 
not  excessive,  but  a  velocity  of  500  ft.  per  sec.,  not  uncommon  in  turbine 
operation,  gives:  SH  =  26,500.  This  does  not  include  the  effect  of  the 
blades,  which,  when  the  radial  thickness  of  the  rim  is  small  compared  to 
the  diameter,  may  be  found  by  dividing  the  centrifugal  force  of  the 
blades  by  2?ra,  where  a  is  the  area  of  the  rim  section  in  sq.  in.  If  we 
consider  a  ring  with  an  axial  thickness  t0,  and  Sc  the  radial  stress  due  to 
some  external  load  such  as  the  blades,  the  resulting  tangential  stress  is: 


2o 

This  is  only  strictly  true  if 
the  radial  thickness  of  the 
ring  is  very  small  relative  to 
DO,  but  as  stated  in  Ch&p. 
XXX,  may  be  assumed  cor- 
rect for  usual  flywheel  pro- 
portions. 

Assume  for  example  that 
Sc  =  16,000,  To  =  1,  and 
D0/2a  =  5.  Then  for  a  rim  velocity  of  500  ft.  per  sec.,  the  tangential 
stress  due  to  rim  load  would  be  80,000  Ib. 

It  is  obvious  that  such  a  stress  is  too  great,  and  the  radial  thickness 
of  the  ring  must  be  increased.  The  strength  does  not  increase,  however, 
in  proportion  to  the  added  area,  and  the  axial  thickness  must  often  be  in- 
creased toward  the  center  of  the  wheel  as  shown  in  Fig.  438. 

The  thickness  to  is  determined  from  the  radial  load  due  to  blading  and 


FIG.  438. 


TURBINE  WHEELS 


651 


rim.     The  thickness  at  any  other  radius,  for  discs  having  a  central  hole, 
is  commonly  found  from  the  exponential  equation : 


(2) 


The  principal  stresses  produced  by  the  centrifugal  force  of  the  blading 
and  by  the  rotation  of  the  disc  itself  .are  radial  and  tangential,  the  latter 
being  the  most  important.  The  effect  of  forcing  the  wheel  upon  the 
shaft  may  probably  be  safely  neglected.  This  produces  an  initial  stress, 
probably  nearly  static,  and  not  increased  by  rotation;  on*  the  other 
hand,  it  probably  decreases  the  range  of  stress  due  to  starting  and 
stopping. 

Then  the  radial  stress  due  to  rim  loading  varies  from  Sc  at  the  radius 
TO  to  zero  at  the  surface  of  the  bore;  and  due  to  the  disc  itself  it  increases 
from  zero  at  radius  r0  to  a  maximum,  and  back  to  zero  at  the  bore.  The 
tangential  stress  is  always  greater  than  zero  and  is  best  shown  by  the 
chart  of  Fig.  440. 

Probably  no  exact  mathematical  analysis  has  been  devised  for  the  case 
of  the  rotating  disc;  those  which  are  used  are  greatly  involved  and  the 
formulas  are  awkward  and  cumbersome,  necessitating  extensive  use 
of  trial  and  error.  After  a  vain  attempt  at  simplification  it  was  decided 
to  use  a  set  of  curves,  the  values  of  which  were  derived  from  another  set 
of  curves  in  Martin's  excellent  work.  The  mathematical  treatment  of 
the  subject  may  be  found  in  treatises  on  the  steam 
turbine  by  Stodola  and  Jude,  and  in  Morley's 
Strength  of  Materials. 

216.  It  may  be  assumed  that  the  blades,  and  a 
portion  of  the  shaded  area  of  the  rim  in  Fig.  439, 
must  be  carried  by  the  disc,  involving  one  or  more 
trial  calculations,  or,  safely,  the  weight  of  the  entire 
rim  maybe  included.  In  either  case  the  rim  weight 
will  be  denoted  by  WR  and  the  weight  of  blades, 
shrouding,  spacers,  etc.  by  WB-  Taking  r0  at  the 
center  of  gravity  of  rim  and  blades  is  entirely 
arbitrary,  but  safe.  It  may  be  taken  at  the  thin- 
nest section  of  the  disc  where  it  joins  the  rim. 

Were  the  rim  revolving  free  it  would  have  the 
stress  SH,  and  a  correspondingly  great  tangential 
strain;  but  the  strain  in  the  rim  must  equal  that  in  the  disc  at  the  point 
of  attachment,  and  the  stress  will  therefore  be  reduced  to  SL.  The  radial 
stress  in  the  rim  may  be  considered  as  zero,  but  in  the  disc  it  has  the  value 


FIG.  439. 


652  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

Sc.     If  S0  is  the  assumed  tangential  stress  at  thife  point,  the  simple 
equivalent  stress  from  Formula  (6),  Chap.  XXI  is: 

S    =  S     -^- 
0       m 

For  equal  strains  in  rim  and  disc : 

SL  _  SE  _     I  /o          Sc 


where  m  is  the  reciprocal  of  Poisson's  ratio. 
Then: 

&  =  So  -  §  (3) 

When  So  is  finally  determined  (it  will  equal  ST), 

m  —  _£ C4) 

rr       o  v*/ 

This  may  be  used  as  a  check  by  comparing  with  values  of  m  in  Table 
79,  Chap.  XXI. 

If  So  =  Sc  and  m  =  3%,  SL  =  Q.7S. 

The  fraction  of  the  rim  carried  by  itself  under  stress  SL  is : 

& 


SH        SH 

The  remainder,  1  —  q,  must  be  carried  by  the  disc.     Substituting  the 
value  of  SH  from  (1)  gives: 

f  . 


107     \100 
Then  the  entire  extraneous  weight  carried  by  the  disc  is: 

W  =  (1  -  q)WR  +  WB  (6) 

From  the  first  statement  of  Formula  (3),  Chap.  XXX,  the  load  per 
inch  of  periphery  is  found  to  be: 


22  V100 
The  required  thickness  at  radius  r0  is  then  : 


UOO 
And  from  (2)  : 


at  any  radius  r. 

The  total  tangential  stress  at  any  radius  is  : 

ST  =  kTSH  +  CTSC  (9) 


TURBINE  WHEELS  653 

And  the  total  radial  stress: 

SK  =  kKSu  +  cRSc  (10) 

where,  as  given  by  (1): 


Stress  ST  is  generally  the  greater  and  is  usually  a  maximum  at  the  bore; 
but  in  some  cases  it  is  well  to  try  for  greater  values  of  r/r0,  especially 
for  larger  values  of  n. 

An  inspection  of  the  general  stress  equations  makes  it  clear  that  for 
similar  discs  the  stress  varies  directly  as  the  square  of  the  peripheral  ve- 
locity (or,  as  D02N2)  ;  then  the  factors  kT,  CT,  kR  and  CR  are  suitable  for 
any  disc,  and  are  given  in  Figs.  440  and  441  for  three  ratios  of  d  to  D0, 
and  for  various  ratios  of  r  to  r0.  Values  of  k  and  c  were  taken  from 
Martin's  curves  for  n  —  0,  1  and  2,  and  a  smooth  curve  drawn  between 
these  points  for  several  values  of  r/r0  ;  other  values  may  be  found  by  inter- 
polation. It  is  probable  that  as  great  accuracy  as  the  problem  demands, 
or  as  is  consistent  with  the  practical  accuracy  of  the  analysis,  may  be 
obtained  by  the  use  of  these  curves. 

In  applying  the  equations  and  the  curves,  solve  for  (5),  (6),  (7)  and 
(11),  and  decide  upon  suitable  values  of  S0  and  Sc.  Take  a  trial  value  of 
n,  select  kT  and  CT  at  the  bore  and  solve  for  ST  in  (9).  This  should  usu- 
ally be  equal  to  Sc,  and  different  values  of  n  should  be  tried  until  the 
right  value  of  ST  is  found.  It  may  be  safe  to  check  the  value  when  r/r0 
is  0.9;  then  ra  may  be  checked  by  (4),  which  will  be  nearly  as  assumed  if 
ST  for  r/r0  =  1  is  practically  equal  to  S0. 

If  all  is  satisfactory,  determine  enough  values  of  t  from  (8)  to  outline 
the  wheel  section. 

For  a  disc  without  a  central  hole,  such  as  the  wheel  of  the  DeLaval 
simple  impulse  turbine,  the  formula  is  : 

i/jLyrA\»_>tt 

t  =  io€sc  \ioo)  lAioJ      \w)  J  (12) 

• 

where  e  =  2.718,  the  base  of  the  Naperian  system  of  logarithms.  Then 
t0  may  be  found  from  (7)  as  before. 

The  stress  is  assumed  to  be  uniform  throughout  the  wheel,  the  tan- 
gential and  radial  stresses  being  equal.  Jude  says  the  reasoning  upon 
which  Formula  (12)  is  based  is  open  to  grave  doubt;  however,  practical 
results  are  obtained  by  its  use. 

If  holes  are  bored  in  a  disc  for  the  purpose  of  equalizing  pressure  on 
the  two  sides,  it  may  be  assumed  that  the  stress  at  tbe  radius  at  which  the 


654  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

OJ 


-  c 


TURBINE    VHEELS 


655 


i 


—  o 


7 


-  C 


656  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

holes  are  bored  is  increased  in  the  same  ratio  that  the  circumferential 
section  is  decreased. 

If  the  rim  is  wide,  as  when  two  or  more  rows  of  blades  are  carried,  it 
is  probable  that  the  stress  at  the  edges  is  greater  than  SL,  though  still  less 
than  SH.  An  extreme  case  is  the  drum  of  a  reaction  turbine  with  web 
connections  to  the  spindle.  The  axial  length  between  the  webs  or  sets 
of  arms  may  be  so  great  as  to  receive  practically  no  support  therefrom, 
the  stress  being  nearly  equal  to  SH  due  to  the  rotation  of  the  drum  itself. 
To  this  must  be  added  the  stress  due  to  the  centrifugal  force  of  the  blad- 
ing,  which  may  be  found  by  dividing  the  centrifugal  force  by  2?ra,  as 
previously  stated.  The  area  a  may  be  taken  as  the  product  of  the  thick- 
ness of  the  drum  and  the  distance  from  center  to  center  of  the  two  adja- 
cent rows  of  moving  blades.  As  the  rim  velocity  of  the  drum  type  is 
usually  less  than  360  ft.  per  sec.,  there  is  little  difficulty  in  keeping  the 
stresses  within  practical  limits. 

217.  Material  for  turbine  wheels  may  range  from  ordinary  open 
hearth  machinery  steel,  to  the  alloy  steels,  and  may  be  selected  from 
Tables  73  to  78,  Chap.  XXI.  Working  stress  may  be  determined  by 
selecting  a  factor  of  safety  from  Par.  159,  Chap.  XXI,  when  the  nature 
of  the  loading  is  determined. 

Changing  the  notation  of  Formula  (141),  Chap.  VI,  and  taking  the  dis- 
tance from  the  center  to  where  the  load  is  applied  in  inches  instead  of 
feet,  gives: 

FrN 
63,000 
or, 

p  =  ?3W  (13) 

where  F  is  the  force  in  Ib.  at  any  radius,  required  to  transmit  H  at  N 
r.p.m.     A  shearing  stress  SD  is  produced  over  the  area, 

a  =  2irrt 
or, 

_  F  _  MyXXKff 

a  ~        rHN 

It  will  be  found  that  SD  is  insignificant  compared  to  ST,  so  that  the 
live  load  due  to  driving  may  be  neglected.  The  stresses  due  to  rotation 
may  be  practically  considered  as  static  stresses  and  a  standard  factor  of 
safety  of  2  employed.  Due  to  possible  slight  vibration,  a  factor  of  judg- 
ment of  from  1.15  to  1.35  may  be  employed,  an  average  giving  a  total 
factor  of  2.5.  In  an  example  of  disc  design  given  by  Martin,  a  working 


TURBINE  WHEELS 


657 


stress  of  16,000  is  assumed  for  mild  steel;  a  factor  of  2.5  gives  an  elastic 
limit  of  40,000  lb.,  which  seems  reasonable. 

For  high-speed  wheels  the  use  of  high  stresses  is  necessary,  and  it  may 
be  possible  that  the  theory  of  rotating  discs  is  more  accurate  with  high 
stresses;  at  any  rate  low  values  of  stress  in  the  formulas  result  in  absurd 
dimensions  which  are  never  found  in  practice.  There  can  be  little  doubt 
that  in  a  rapidly  increasing  thickness  of  metal  the  stress  distribution  is 
not  that  indicated  by  any  theory  that  has  any  semblance  of  simplicity. 
However,  judged  by  the  usual  standards  of  machine  design,  the  methods 
used  to  determine  the  dimensions  of  steam  tur- 
bine parts  must  be  satisfactory  when  it  is  con- 
sidered that  proportionate  to  the  period  of  its 
practical  application,  no  other  type  of  prime 
mover  has  probably  had  so  few  accidents. 

218.  Blading  Design. — The  process  of  de- 
termining blade  angles  and  lengths  was  discussed 
in  Chap.  XV.  The  determination  of  width  is 
rather  arbitrary  and  depends  upon  the  length. 
The  matter  of  angles  and  width  being  fixed,  the 
radius  and  location  of  center  may  be  found. 
The  known  quantities  in  Fig.  442  are  the  angles 
A  and  B,  and  the  width  a  +  b.  It  is  seen  from  the  figure  that:  a  =  R 
cos  A,  and  b  —  R  cos  B.  Then: 

a  _  cos  A 
b  = 


r 


FIG.  442. 


or, 


Then: 


Or: 


And 


cos  B 


cos  A 
-  5 
cos  B 


cos 


b  = 


a  +  b 


1  + 


cos  A 
cos  B 


R  = 


cos  B 


(15) 


(16) 


The  blade  pitch  and  form  of  the  back  depend  upon  the  radius  r. 
As  stated  in  Chap.  XV,  as  large  a  pitch  as  practicable  probably  tends  to 
reduce  friction,  and  this  is  obtained  by  making  r  small.  In  impulse 

42 


658 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


blading  the  center  of  radius  r  usually  lies  within  the  blade  section,  some- 
times on  the  curve  of  radius  R.  In  Fig.  443,  p  is  the  maximum  pitch 
which  will — at  least  theoretically — direct  the  discharge  steam  at  the 
angle  B  relative  to  the  blade.  It  may  be  easily  shown  that: 

A  +B 


cos 

p  = ^_ 

x  cos  B 


Or: 


cos 


A  +  B 


p  =  x 


cos  B 


(17) 


FIG.  443. 


Good  results  will  usually  be  obtained  when  x  =  R\  then: 


cos 


A  +  B 


b-  cos 


A  +  B 


p  = 


cos  B 


cos2  B 


a  +  b 


2  cos  B-  cos 


A  +  B 


(18) 


Formula  (18)  may  be  used  to  find  the  pitch  tentatively.  If  all  spaces 
are  to  be  equal  a  slight  change  may  be  necessary.  There  is  no  definite 
rule  for  spacing  and  it  is  probable  that  more  or  less  latitude  is  permissible. 


TURBINE  WHEELS  659 

Strength  of  Blades. — Formula  (24),  Chap.  XV,  gives  the  tangential 
force  fw  which  must  be  divided  among  the  blades  taking  steam  at 
one  time.  For  full  peripheral  admission  this  would  be  all  the  blades  on 
a  wheel.  Formula  (25)  Chap.  XV,  gives  the  axial  force  fF)  this  being 
usually  small.  If  I  is  the  distance  from  the  center  of  the  blade  to  the 
weakest  section  in  inches,  and  n  is  the  number  of  blades  receiving  steam, 
the  bending  moment  is: 

M  =  f-  -  I  (19) 

n 

where  /  refers  to  either  fw  or  fF.     If  Z  is  the  section  modulus  of  the  section 
considered  and  S  is  the  stress: 

&  =  ~  (20) 

Should  the  section  be  irregular,  as  the  blade  section,  Z  may  be  found 
by  the  graphical  method  given  in  Appendix  1.  Stress  of  the  same 
sign  (either  tensile  or  c6mpressive)  may  be  determined  by  (20)  at  point 
of  maximum  stress  by  using  both  fw  and  fF,  which  act  at  right  angles  to 
each  other;  the  sum  will  be  the  maximum  stress. 

Reducing  Sc  by  making  t0  greater,  reduces  the  value  of  n,  but  the  ratio, 


will  give  greater  values  of  t  at  all  values  of  r,  so  that  nothing  is  gained. 

If  certain  values  of  t  are  desired  at  hub  and  rim,  n  may  be  found  from 
(8),  Sc  from  (7)  and  ST  from  (9).  Factor  of  safety  and  material  may 
then  be  chosen.  Unless  an  elaborate  mathematical  analysis  is  used,  it  is 
probably  safer  to  cause  intermediate  values  of  t  to  lie  on  the  curve  plotted 
from  (8) .  This  is  one  of  the  cases  where  the  addition  of  metal  may  be  a 
source  of  weakness  rather  than  strength. 

219.  Application  of  Formulas. — Assume  a  wheel  similar  to  Fig.  444. 
Let  the  weight  of  the  blades  be  35  Ib.  (WB)  and  the  total  weight  of  the  rim 
74  Ib.  (WK).  Also  let  N  =  5000,  D0  =  25.125  in.  and  r?  =  12.562  in. 
Let  the  material  be  nickel  steel  with  an  elastic  limit  of  50,000  Ib.;  then 
with  a  factor  of  safety  of  2.5,  S0  =  Sc  =  20,000.  Then  from  (5),  q  = 
0.445.  From  (6),  W  =  76  and  from  (7),  to  =  0.73. 

Assume  that  d  =  0.2D0.  From  (1),  SH  =  31,500.  By  several  trials, 
solving  for  stress  at  diameter  d,  n  was  taken  as  1.5.  Then  (9)  and  the 
charts  give: 

ST  =  (0.355  X  31,500)  +  (0.425  X  20,000)  =  19,700. 

For  r/r0  =  0.9,  ST  =  17,510  and  for  r/r0  =  1,  ST  =  18,380.  At  the 
outside  of  the  hub  where  r/r0  =  0.24,  ST  =  18,400.  As  the  hub  strength- 


660 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


ens  the  disc  in  this  region,  n  might  have  been  less.  The  thickness  at  the 
hub  where  r  =  3.5  in.  is  given  by  (8),  and  is  3.68  in.  Values  of  t  for 
various  sections  may  be  found  by  (8)  and  the  curve  drawn. 

Taking  S0  =  ST,  and  SL  =  0.7  S0,  (4)  gives:  m  =  3.63.  This  is  some 
larger  than  the  value  assumed,  and  the  value  of  SL  was  based  upon  the 
other  value,  but  the  result  is  probably  as  nearly  correct  as  the  method  in 
general. 

220.  Design  from  Practice. — Wheels  are  made  of  various  forms,  from 
a  disc  of  uniform  thickness  to  the  disc  of  uniform  strength  of  the  DeLaval 


Defiail  of  Rim 
FIG.  444. — Southwark-Rateau  turbine  wheel. 


Section  B~B 


Class  A  turbine.  The  latter  is  designed  by  Formulas  (7)  and  (12).  A 
disc  of  uniform  thickness  is  expressed  by  (2)  when  n  =  0.  In  some  cases 
t0  is  found  from  (7);  then  t  at  the  hub  from  (8),  so  that  ST  from  (9)  is 
within  limits;  then  a  straight  line  connects  these  two  dimensions,  making 
a  disc  easy  to  machine. 

Figure  444  is  a  wheel  designed  to  run  5000  r.p.m.  by  the  Southwark 
Foundry  and  Machine  Co.,  Philadelphia,  Pa.  The  curve  of  this  wheel 
does  not  check  with  (8) ;  an  approximation  was  probably  made  to  simplify 
construction. 


TURBINE  WHEELS 


661 


— 


FIG.  445. — DeLaval  Class  C  turbine  wheel  fastening. 


FIG.  446. — DeLaval  multi-stage  turbine  wheel  fastening. 


662 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Figure  445  shows  the  method  of  attaching  the  wheel  of  the  DeLaval 
velocity-stage  turbine  to  the  shaft.  Each  wheel  is  held  by  a  Woodruff 
key,  and  the  wheels  are  spaced  and  centered  by  rings.  All  are  held  against 


-609 


FIG.  447. — Kerr  turbine  wheel  fastening.        FIG.  448. — Rim  of  Curtis  turbine  wheel. 

a  shoulder  on  the  shaft  by  a  nut.  Fig.  446  shows  the  method  of  attaching 
the  wheels  of  the  DeLaval  pressure-stage  turbine  to  the  shaft.  Tapered 
sleeves  are  keyed  to  the  shaft,  the  wheels  being  drawn  onto  these  by  a  nut. 


FIG.  449. — Ridgway  turbine  blading. 

The    labyrinth    packing    between    the    diaphragms    and  wheel  is  also 
shown. 

Figure  447  shows  the  method  of  fastening  the  disc  to  the  turbine  shaft 


TURBINE  WHEELS 


663 


used  on  the  Kerr  Economy  turbine  of  the  smaller  sizes.  The  hubs  are 
prevented  from  turning  by  dowels.  The  discs  are  of  uniform  thickness. 
In  the  larger  sizes  the  wheels  are  made  in  one  piece  and  held  to  the  shaft 
by  keys. 

A  section  of  the  rim  of  a  Curtis  turbine  wheel,  built  by  the  General 
Electric  Co.,  is  shown  in  Fig.  448.  This  is  a  2- velocity  stage  wheel  used 
in  the  first  pressure  stage.  The  blades  shown  are  drop-forged  with  dis- 
tance piece  integral  with  blade.  A  projection  on  the  blade  passes  through 
a  shroud  ring  and  is  riveted.  This  strengthens  the  blading  and  forms 
one  side  of  the  steam  passage  through  the  blades. 


FIG.  450. — Ridgway  turbine  blade. 

The  method  of  fastening  the  blading  of  the  Ridgway-Rateau-Smoot 
turbine  is  shown  in  Fig.  449,  and  a  detail  of  the  blade  in  Fig.  450.  In  the 
Ridgway  blades  no  fillers  or  shrouds  are  used,  as  these  are  formed  by  the 
blades  themselves.  These  blades  are  machined  all  over  and  usually 
made  of  monel  metal.  An  advantage  of  the  bulb-end  type  of  blades  is 
that  in  case  of  accident  to  any  of  the  blades,  any  one  may  be  removed 
without  disturbing  the  rest. 

References 

Dynamic  balance A.S.M.E.  Journal,  August,  1916. 

Blade  fastening  for  steam  turbines Power,  July  20,  1915. 

Strength  on  materials Prof.  Arthur  Morley. 

Design  and  construction  of  steam  turbines.  .H.  M.  Martin. 


CHAPTER  XXXII 

TURBINE  SHAFTS 
Notation. 

d  =  diameter  of  shaft  in  inches. 
di  =  inside  diameter  of  hollow  shaft. 
dB  =  diameter  of  journal. 

I  =  length  in  inches  from  center  to  center  of  bearings. 
x  =  distance  to  any  point  on  shaft  in  inches;  used  in  derivation  of 

formulas. 

a  =  length  of  section  of  shaft  to  be  taken  as  isolated  load  =  q-Ax. 
Ax  =  actual  measurement  in  inches  on  bending  moment  diagram, 

corresponding  to  a. 

h  —  pole  distance  in  inches  on  vector  diagram  for  bending  moments. 
k  =  same  for  deflection  diagram. 
z  =  actual   measurement   in   inches   corresponding   to    bending 

moment  on  diagram. 
y  =  same  for  deflection. 
p  =  weight  scale  =  pounds  per  inch. 
q  =  distance  scale  =  inches  per  inch. 
m  =  units  of  z-Ax/d*  per  inch. 
5  =  actual  deflection  in  inches  under  any  load. 
W  =  weight  of  isolated  load  in  pounds. 
w  =  weight  per  inch  of  length. 
S  =  shearing  stress  in  journal  due  to  torque. 
E  =  modulus  of  elasticity. 
/  =  moment  of  inertia  of  shaft  section. 
M  =  bending  moment  at  any  point. 
MM  =  mean  bending  moment  in  length  a  of  shaft,  or  in  length  Ax 

of  diagram. 

N  =  normal  r.p.m.  of  turbine. 
Nc  =  r.p.m.  at  critical  speed.' 
co  =  angular  velocity  at  critical  speed,  in  radians. 
V  =  linear  velocity  in  feet  per  second  of  isolated  load  W  revolv- 
ing at  radius  5. 
H  =  horsepower. 

664 


TURBINE  SHAFTS  665 

221.  Nature  of  the  Problem.—  The  rotative  speed  of  the  reciprocating- 
engine  shaft  is  usually  so  low  that  vibration  needs  little  or  no  considera- 
tion. In  small  gasoline  engines  with  very  high  rotative  speed  the  shaft 
is  so  well  supported  by  bearings  that  vibration  is  not  usually  considered. 
With  the  shaft  of  the  reaction  turbine  the  drum  construction  is  such  that 
the  required  stiffness  is  attained  easily  for  the  speeds  at  which  this  type 
is  operated.  For  the  multi-stage  impulse  turbine  with  disc  wheels,  the 
shaft  receives  comparatively  little  stiffening  from  the  hubs  —  and  this  is 
neglected  in  calculation  —  so  that  the  shaft  must  maintain  stability  by  its 
own  stiffness.  It  must  not  run  at  a  speed  anywhere  near  its  critical 
speed. 

The  determination  of  the  shaft  diameter  is  therefore  an  entirely  differ- 
ent problem  from  that  of  the  engine  shaft,  the  latter  being  one  of  bending 
and  twisting.  These  are  both  insignificant  in  the  turbine  shaft  except  at 
or  near  a  certain  speed  —  the  critical  speed. 

For  preliminary  calculations  it  is  convenient  to  have  some  formula  to 
assist  in  fixing  dimensions;  then  these  may  be  checked  by  the  methods  of 
the  following  paragraphs. 

If  the  shaft  be  assumed  to  transmit  power,  the  smallest  diameter, 
which  may  be  taken  at  the  journal,  will  be  given  by  the  formula: 


where  H  is  the  horsepower,  S  the  shearing  stress  and  N  the  r.p.m.  The 
stress  S  must  be  taken  low  —  2000  to  2500  —  to  give  practical  results. 
Length  of  bearing  may  be  determined  from  Par.  52,  Chap.  XI,  and  any 
necessary  compromises  made. 

The  shaft  between  the  bearings  is  larger,  sometimes  having  a  number 
of  different  diameters  increasing  toward  the  center. 

A  tentative  determination  of  an  equivalent  straight-sided  shaft  may 
be  made  by  assuming  a  uniform  load  composed  of  all  the  wheels  and  a 
shaft  of  a  greater  diameter  than  that  given  by  (1)  ;  then  by  the  general 
principles  of  Par.  222  the  following  formula  may  be  derived: 

1850    lEI       1850    \EIl  /0, 

Nc"  ~:  -- 


where  E  is  the  modulus  of  elasticity,  /  the  moment  of  inertia,  w  the  load 
per  in.  of  length  including  shaft,  W  the  total  weight  in  Ib.  and  I  the 
length  of  shaft  in  inches  between  bearing  centers.  For  a  solid  shaft, 
taking  #  =  30,000,000: 

Nc  =  2,250,000        2  (3) 


666  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

NO  should  be  much  greater  or  much  less  than  the  actual  shaft  speed,  N. 
By  assuming  the  number  of  r.p.m.,  (2)  may  be  written: 


(4) 

JH-7CFV/.LV 

Or  from  (3) 

*-   l 


1490 

As  actual  conditions  are  usually  not  so  favorable  to  stiffness,  the 
actual  critical  speed  will  probably  be  less  than  that  given  by  (2)  or  (3) . 

A  good  deal  of  information  about  the  principles  underlying  critical 
speed  maybe  found  in  Morley's  Strength  of  Materials,  to  which  the  author 
is  largely  indebted  for  the  ideas  of  the  next  two  paragraphs.  An  abso- 
lutely correct  solution  is  probably  impossible  from  the  practical  stand- 
point, due  to  various  influences  which  may  not  easily  be  taken  into 
account,  but  a  method  will  be  given  which  is  accurate  in  principle  to 
within  narrow  limits;  if  this  is  carefully  applied,  practical  results  may  be 
expected. 

222.  Critical  Speed. — If  a  body  is  acted  upon  by  a  periodic  force 
having  the  same  frequency  as  the  natural  vibration  of  the  body,  the  total 
energy  will  be  increased  at  each  application,  increasing  the  strain  energy 
and  therefore  the  stress.  If  this  continues  until  the  elastic  limit  of  the 
material  is  reached,  distortion  of  the  body  will  result.  The  frequency 
causing  this  is  called  the  critical  frequency. 

When  a  body  vibrates,  the  energy  is  of  two  kinds,  kinetic  and  poten- 
tial, their  sum  being  constant.  In  the  mean  position  the  velocity  is  a 
maximum  and  the  energy  is  all  kinetic;  at  the  extremes  it  is  all  potential. 

The  motion  of  natural  vibration  is  simple  harmonic,  and  therefore 
displacements  may  be  treated  as  projections  on  the  plane  of  vibration,  of 
a  body  revolving  with  a  constant  angular  velocity. 

Assume  the  body  in  question  to  be  a  turbine  shaft,  deflected  under  its 
own  weight  and  that  of  the  disc  wheels;  with  carefully  balanced  shafts 
this  weight  may  be  considered  as  the  periodic  force,  the  period  being  that 
of  rotation.  The  force  acts  in  but  one  direction,  but  the  rotation  of  the 
shaft  presents  opposite  sides  alternately  to  the  force,  producing  the  same 
effect  as  forced  vibration.  If  the  rotative  speed  is  the  same  as  the  natu- 
ral period  of  vibration  it  is  called  the  critical  speed;  then  the  potential 
energy  required  to  force  the  shaft  to  its  deflected  position  is  equal  to  the 
kinetic  energy  of  the  entire  mass,  should  its  axis  whirl  on  a  surface  of 
revolution  developed  by  the  center  of  the  deflected  shaft,  the  speed  of, 
whirling  being  the  critical  speed. 


TURBINE  SHAFTS  667 

If  the  system  is  not  in  perfect  balance  there  is  a  tendency  to  whirl  at 
all  speeds  below  the  critical  speed,  the  latter  probably  being  lower  on 
this  account.  As  the  speed  is  increased  beyond  this  point  the  tendency 
to  whirl  is  reduced,  and  the  shaft  tends  to  revolve  around  the  axis  of 
gravity  of  the  system. 

It  is  more  convenient,  and  sufficiently  accurate  for  practical  purposes 
to  assume  each  wheel  or  portion  of  shaft  as  an  isolated  load.  Let  these 
loads  be  TFi,  W2,  etc.,  and  the  deflections  under  the  loads  5i,  S2,  etc.  re- 
spectively. These  deflections  are  radii  of  the  circles  the  loads  are  as- 
sumed to  revolve  upon  at  the  instant  the  critical  speed  is  reached.  The 
potential  energy  of  these  loads  in  ft.-lb.  is: 


The  kinetic  energy  is  : 


900  X  2880 
Equating  (6)  and  (7)  gives: 

T     *»C  • 


2880 

2  (W S2)  (7) 


900  X  2880  24 

From  which: 


If  in  (8),  load  w  per  unit  length  were  used  in  place  of  W,  the  expres- 
sion would  be: 


Then  from  the  equation  of  the  elastic  curve,  if  the  value  of  d  in  terms 
of  x  be  substituted,  the  result  for  a  shaft  of  uniform  section,  with  uni- 
form load,  and  for  ends  supported,  would  be  given  by  Formula  (2) . 

223.  Deflection. — Before  Formula  (8)  can  be  applied  it  is  necessary  to 
determine  the  deflection  under  each  load  W;  this  may  best  be  done  graphi- 
cally. The  underlying  principles  of  what  follows  are  thoroughly  treated 
in  Morley's  Strength  of  Materials,  but  an  attempt  will  be  made  to  ex- 
plain their  practical  use  in  such  a  way  that  they  may  be  applied  if  this 
fundamental  knowledge  is  lacking. 


668  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  general  equation  of  the  elastic  curve,  when  the  curvature  is  slight, 
is: 

dx2==Kl  (9) 

Integrating  gives: 

dv      CM 

i-jm          :       ™ 

This  gives  the  slope  at  a  certain  point  in  the  deflection  curve;  another 
integration  gives  y,  the  deflection  from  some  reference  point. 
Another  fundamental  beam  equation  is : 

<PM 


where  M  is  the  bending  moment  in  in.-lb.  and  w  is  the  load  per  inch  of 
length.  Integrating  (11)  gives: 

dM        r 

-fr   "=  J  W'dx  (12) 

This  gives  the  slope  at  a  certain  point  of  the  bending  moment  curve, 
and  another  integration  gives  the  bending  moment  when  certain  condi- 
tions are  known.  Comparing  (9)  and  (10)  with  (11)  and  (12),  it  is  obvi- 
ous that  if  a  graphical  method  may  be  used  to  find  M  when  the  load 
curve  (curve  of  w)  is  known,  the  same  method  may  be  used  to  find  y 
when  the  curve  of  M/EI  is  known. 

If  the  bending  moment  due  to  a  continuous  load — whether  uniform  or 
varying — is  to  be  found  graphically  it  is  divided  up  into  sections,  each  one 
of  which  is  assumed  equivalent  to  an  isolated  load  located  at  the  center  of 
gravity  of  the  section  of  continuous  load.  Let  W  be  one  of  these  equiva- 
lent isolated  loads  between  x2  and  x\ ;  then : 


W  =  f"wdx  (13) 


the  quantity  in  the  integral  sign  being  the  area  of  the  load  curve  between 
xz  and  x\. 

Likewise  the  quantity: 


r 

Jxi 


is  the  area  of  the  bending  moment  curve  between  x2  and  x\.     If  MM  is 
the  mean  height  of  this  area  and  a  the  length  to  certain  scales, 

MMa        Cx*  M 


TURBINE  SHAFTS 


669 


Then  from  (10),  (12),  (13)  and  (14),  if  W  is  used  for  plotting  the  bend- 
ing moment  curve, 

MMa 
El 

may  be  used  for  plotting  the  curve  of  deflection. 

An  important  matter  is  the  fixing  of  the  scales,  which  may  be  taken 
as  follows: 

Let  1  in.  =  p  Ib.     ( =  p  units  of  W). 

Let  1  in.  =  q  in.     (of  measurement  along  shaft). 

Before  determining  other  scales  the  bending  moment  diagram  will 
be  plotted  in  Fig.  451.  The  loads,  Wi,  W2,  etc.,  may  vary  in  amount  and 
be  distributed  in  any  way.  Lay  these  loads  to  any  convenient  scale 


FIG.  451. 

along  the  vertical  line  consecutively  as  shown  in  Fig.  4515.  Take  a  pole 
at  any  point  0,  drawing  rays  1,  2,  3,  etc.,  to  join  the  loads  as  shown. 
Starting  at  any  point  at  the  left  reaction,  draw  link  1  parallel  to  ray  1 
until  it  cuts  the  load  W;  draw  link  2  parallel  to  ray  2,  and  so  on  until 
the  right  reaction  is  reached.  Connect  the  two  ends.  A  line  drawn 
from  pole  O  parallel  to  this'  determines  the  reactions,  which  may  be 
used  in  determining  bearing  pressures. 

The  measurement  z  on  A  is  distance,  so  the  scale  q  must  be  applied 
to  it;  the  measurement  h  is  on  the  force  diagram  and  scale  p  must  be  used. 
Let  the  quantities  h  and  z  be  actual  inches  on  the  diagram.  Then  the 
bending  moment  at  any  point  at  which  z  is  measured  is: 


in  which  pqh  is  a  constant. 


M  =  pqhz 


(15) 


070 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


In  (14),  E  is  constant;  /  may  be  constant,  or  in  stepped  shafts  it 
varies.  It  will  be  treated  as  variable  in  order  to  be  more  general.  To 
simplify  explanation  it  will  first  be  assumed  that  the  wheel  is  located  at 
the  center  of  each  step,  the  lengths  of  which  are  a\,  a2,  etc.  In  this  case 
the  weight  of  the  wheel  and  the  step  are  equal  to  W.  The  moment  dia- 
gram in  Fig.  452  was  drawn  as  in  Fig.  451.  The  quantity  in  (14),  which 
is  analogous  to  W  may  be  reduced  to  constants  and  variables;  the  latter 
only  need  be  plotted.  Taking  the  value  of  M  from  (15) : 

MMa  _  pqhz  q>Ax 

^T  ''       ~Ef~ 


(16) 


Mrftb 


K-a,->  <-  ar^--ffj-f  r-a4->  t-af 


+  t-eff.++a.. 


FIG.  452. 


The  quantity  Az  is  in  actual  inches  on  the  diagram. 
quantities  in  brackets  are  variable.     For  solid  shafts: 


In  (16)  the 


64 

where  d  is  the  diameter  of  the  shaft  in  inches.  Should  the  shaft  be  hol- 
low, d4  may  be  replaced  by  d4  —  di4,  where  d±  is  the  inside  diameter.  As 
most  shafts  are  solid,  (16)  becomes: 

MMa  _  64pqzh/ 

~wr     *E  v 


(17) 


The  quantity  in  brackets  may  be  plotted.     As  the  product  z-Ax  is 
small,  it  may  be  better  to  include  part  of  the  constants  in  the  brackets  in 


TURBINE  SHAFTS  671 

some  cases,  especially  if  d  is  large.  With  a  shaft  of  uniform  diameter 
between  bearings,  d  is  constant  and  may  be  taken  out  of  the  brackets. 
When  it  is  decided  what  shall  be  plotted,  the  quantities  may  be  laid 
off  in  succession  as  shown  in  Fig.  452C ;  a  pole  may  be  located  at  any  point 
as  shown,  the  rays  drawn  and  the  corresponding  links  laid  off  parallel 
to  them  on  B.  The  quantities  y  and  k  are  in  actual  inches  on  the  dia- 
grams; the  former  must  be  multiplied  by  the  distance  scale  q  and  the 
latter  by  the  scale  for  the  quantity  plotted ;  or : 

.  .  .,      ,  z-Az 

1  in.  =  m  units  of  —n- 
d4 

In  addition  to  this  the  constants  of  (17)  must  be  used,  as  the  entire 
quantity: 

MMa 
El 

must  be  used  to  determine  the  deflection  d.     The  vertical  intercepts  of 
Fig.  4525  are  proportional  to  the  deflection;  then: 

''"  '  i.y^.y-Ky  '     *  :       (18) 

Should  the  wheels  have  offset  hubs,  and  not  be  located  at  the  centers  of 
the  steps,  separate  moment  diagrams  may  be  drawn,  one  for  the  shaft 
and  one  for  the  wheels,  taking  the  loads  at  the  center  of  gravity  of  wheel 
and  step  in  each  case.  Should  the  steps  be  long  they  may  be  divided  into 
two  or  more  loads.  These  two  diagrams  may  be  combined ;  then  dividing 
into  suitable  lengths,  each  one  of  which  covers  a  portion  of  the  shaft  of 
constant  diameter,  the  deflection  curve  may  be  plotted  and  the  deflection 
under  each  load  found.  The  shaft  steps  and  wheels  may  be  kept  separate 
if  desired  in  determining  S(TFS)  and  2(W d2). 

The  deflection  curve  may  be  drawn  more  correctly  by  an  inscribed 
curve  touching  the  sides  of  Fig.  4525. 

The  dimensions  h  and  k  should  be  chosen  so  that  the  moment  and  de- 
flection curves  will  not  be  too  flat,  as  it  is  easier  to  draw  a  deeper  dia- 
gram accurately. 

224.  Application  of  Formulas. — As  an  example  a  13-stage  turbine  has 
been  taken.  The  data  was  partly  assumed  and  not  very  accurate. 
The  diagrams  were  similar  to  Figs.  451  and  452  and  will  not  be  repeated. 
In  plotting  the  deflection  curve,  a  was  taken  instead  of  Az;  this  re- 
moved one  q  from  the  constant  in  (18).  The  scales  were:  q  =  6;  p  = 
300;  m  =  0.04.  Also:  h  =  3  and  k  =  3.  From  (18),  K  =  0.00264. 
The  length  from  center  to  center  of  bearing  is  67  in.  and  the  total 
weight  1850  Ib.  The  remainder  of  the  data  is  placed  in  Table  102. 


672  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

TABLE  102 


No. 

w 

a 

d 

z 

az 
d* 

y 

d 

W8 

TFS2 

1 

80 

10.80 

4.30 

1.42 

0.0440 

1.44 

0.0038 

0.304 

0.00116 

2 

170 

5.12 

4.40 

2.49 

0.0220 

2.20 

0.0056 

0.950 

0.00532 

3 

100 

2.80 

5.30 

3.40 

0.0117 

2.57 

0.0068 

0.680 

0.00463 

4 

100 

2.97 

5.52 

3.33 

0.0106 

2.74 

0.0072 

0.720 

0.00520 

5 

145 

2.97 

5.67 

3.64 

0.0104 

2.92 

0.0077 

1.120 

0.00863 

6 

145 

2.97 

5.80 

3.83 

0.0100 

3.02 

0.0079 

1.145 

0.00905 

7 

145 

2.97 

5.93 

3.94 

0.0094 

3.10 

0.0082 

.188 

0.00975 

8 

145 

2.97 

5.93 

3.96 

0.0095 

3.12 

0.0084 

.220 

0.01025 

9 

145 

3.24 

5.80 

3.90 

0.0112 

3.11 

0.0083 

.205 

0.01000 

10 

175 

3.91 

5.40 

3.73 

0.0172 

3.04 

0.0080 

.400 

0.01120 

11 

200 

4.60 

5.27 

3.43 

0.0204 

2.86 

0.0075 

.500 

0.01125 

12 

250 

5.40 

4.85 

2.82 

0.0275 

2.48 

0.0065 

1.622 

0.01060 

13 

80 

10.80 

4.30 

1.52 

0.0480 

1.48 

0.0039 

0.312 

0.00111 

Taking  S(TF6)  and  S(TF52),  Formula  (8)  gives  for  the  critical  speed: 


If  an  equivalent  diameter  of  5J^  in.  be  assumed  for  a  straight  shaft, 
and  the  total  weight  taken  as  a  uniform  load,  (3)  gives:  Nc  —  2800. 

The  determination  of  Nc  by  either  (8)  or  (3)  should  be  greatly  different 
from  the  actual  turbine  speed  for  safety  —  probably  at  least  100  per  cent. 
greater,  or  50  per  cent,  as  great.  It  is  practically  impossible  to  consider 
all  factors  which  enter  into  the  problem,  but  with  due  allowance  the 
method  may  be  considered  safe  if  the  shaft  is  taken  as  a  beam  supported 
at  the  ends,  and  only  loads  between  supports  be  considered. 

Jude  says  that  "the  chief  concern  is  whether  the  critical  speed  is 
(e.g.)  300  or  3000  r.p.m.;  not  whether  it  is  300  or  310." 

Martin  says  it  is  not  uncommon  to  run  turbines  of  the  disc  type  above 
the  critical  speed.  "In  that  case,  however,  the  efficiency  of  the  turbine 
may  be  expected  to  diminish  with  time,  since  on  every  occasion  on  which 
the  turbine  is  started  up  or  stopped,  the  rotor  has  to  pass  through  its 
critical  speed  and  the  consequent  vibration  gradually  enlarges  the  fine 
clearances  used  where  the  shaft  passes  through  the  high-pressure 
diaphragms." 

Jude  further  says:  "It  is  found  that  if  the  speed  be  increased  quickly, 
so  that  the  critical  velocity  is  of  only  passing  influence,  the  whirl  quiets 
down  —  so  much  so,  in  fact,  that  the  stability  of  the  system  is  in  general 
greater  than  at  speeds  below  the  critical."  He  also  states  that  the  normal 


TURBINE  SHAFTS  673 

speed  should  be  as  remote  from  the  critical  speed  as  possible,  and  that 
properly  it  should  be  above  rather  than  below. 

In  the  single-stage  DeLaval  turbine  the  speed  is  far  above  the  critical 
speed ;  concerning  the  multi-stage  turbine,  however,  it  is  said  in  Catalogue 
D:  "The  use  in  a  multi-stage  turbine  of  a  shaft  running  above  or  near 
its  critical  speed  is  an  error,  not  only  on  the  grounds  of  safety  and  freedom 
from  vibration,  but  also  because  any  whipping  or  eccentric  rotation  of  the 
shaft  will  require  enlarged  clearances  where  the  shaft  passes  through  the 
diaphragms.  In  other  words,  the  leakage  areas  will  need  to  be  consider- 
ably increased.  It  therefore  follows  that  the  total  leakage  is  reduced  by 
using  a  shaft  sufficiently  large  and  stiff  to  suppress  such  vibration  entirely, 
thereby  permitting  the  radial  clearance  to  be  correspondingly  reduced, 
although  the  larger  shaft  has  a  greater  circumference." 

The  use  of  material  for  wheels  which  permit  the  use  of  high  stresses, 
will  both  lighten  the  hub  and  shorten  the  fit,  making  possible  a  shorter 
shaft.  A  straight  shaft  is  stiffer  than  a  stepped  shaft  of  the  same  diam- 
eter at  center,  so  if  some  method  of  mounting  wheels  be  used  which 
permits  a  straight  shaft,  it  will  increase  the  critical  speed. 

For  turbines  of  the  drum  type  the  shafts  are  stiffened  by  the  drum,  and 
the  critical  speed  is  probably  never  reached.  But  high  speeds  are  not 
possible  with  this  type  as  may  be  seen  from  Chap.  XXXI. 

References 

The   design  and  construction  of  steam  turbines H.  M.  Martin. 

Strength  of  materials Prof.  Arthur  Morley. 


CHAPTER  XXXIII 
TURBINE  CASINGS  AND  DETAILS 

225.  There  is  very  little  calculation  involved  in  the  design  of  casings, 
which  are  practically  the  frames  of  turbines;  therefore  this  chapter  will 
be  comprised  of  illustrations  and  descriptions  of  some  of  the  details  of 
several  makes  of  turbine. 

One  of  the  principal  problems  in  connection  with  casings  is  to  provide 
suitable  clearances  for  uneven  expansion  of  casing  and  rotor.  This  be- 
comes more  of  a  problem  in  large  turbines,  so  it  has  been  customary  to 
divide  the  turbine  into  high-pressure  and  low-pressure  elements,  some- 
times on  separate  shafts,  forming  cross-compound  turbines,  and  some- 
times on  the  same  shaft  as  tandem-compound  turbines.  Recently  very 
large  turbines  have  been  placed  in  a  single  casing.  An  accident  to  such  a 
turbine,  doubtless  due  to  expansion,  is  described  in  Power,  March  19, 
1918. 

Some  idea  of  turbine  casings  may  be  obtained  from  the  illustrations 
of  Chap.  IV,  and  these  may  supplement  those  of  this  chapter.  But  few 
dimensioned  drawings  are  given ;  the  treatment  is  therefore  more  quali- 
tative than  quantitative.  In  some  cases  assembly  drawings  are  given, 
the  relations  of  the  parts  being  better  seen  from  these.  No  special  order 
is  adhered  to,  the  parts  of  a  given  make  being  given  together. 

226.  DeLaval  Turbine. — A  general  view  of  the  casing  of  the  Class  B 
turbine  is  shown  in  Fig.  453.     The  lettered  parts  are  known  as: 

A.  Wheel  case. 

B.  Wheel  case  cover. 

C.  Turbine  wheel. 

D.  Inner  packing  bushing  bracket. 

E.  Inner  packing  bushing. 

F.  Outer  packing  bushing  bracket. 

G.  Outer  packing  bushing. 
H.  Outer  bearing  bracket. 
K.  Insulating  cover. 

L.  Exhaust  flange. 
M.  Governor. 
V.  Turbine  shaft. 

Figure  454  shows  a  partial  section  of  a  velocity-stage  Class  C  turbine. 
The  steel  retaining  ring  completely  covers  the  wheels  and  serves  to  protect 
the  cast  iron  casing  in  case  of  accident  to  the  wheels. 

674 


TURBINE  CASINGS  AND  DETAILS 


675 


Figure  455  shows  the  nozzles  for  this  turbine,  A  being  for  the  high- 
pressure  condensing  turbine,  therefore  having  a  great  divergence;  B  being 


FIG.  453. —  DeLaval  Class  B  turbine  casing. 


FIG.  .  454. — Section  of  DeLaval  Class  C  turbine. 


for  low-pressure  condensing,  or  high-pressure   noncoudensing  turbines, 
has  less  divergence. 


676  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


FIG.  455. — Nozzles  for  DeLaval  Class  C  turbine. 


FIG.  456.— Outboard  bearing  bracket  for  DeLaval  Class  C  turbine. 


I 
FIG.  457. — Radial  and  thrust  bearing  of  DeLaval  Class  C  turbine, 


TURBINE  CASINGS  AND  DETAILS 


677 


FIG,  458. — Retaining  rings  of  DeLaval  multi-stage  turbine. 


FIG.  459. — Carbon  packing  of  DeLaval  multi-stage  turbine. 


678 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


Figure  456  shows  a  sectional  elevation  of  the  outboard  bearing  bracket 
and  flexible  coupling  of  the  Class  C  turbine.  The  bearings  are  ring-oiling; 
they  are  entirely  separate  from  the  stuffing  boxes,  so  that  no  steam  or 
water  can  reach  them  or  enter  the  oil  reservoir. 

Figure  457  shows  the  combined  radial  and  thrust  bearing  and  governor 
drive  of  the  DeLaval  velocity-stage  turbine. 

Figure  458  shows  the  steel  retaining  rings  of  the  DeLaval  pressure- 
stage  turbine. 


FIG.  460. — Clearances  of  Ridgway  turbine. 

The  carbon  packing  at  the  low-pressure  end  of  the  DeLaval  pressure- 
stage  turbine  is  shown  in  Fig.  459.  Provision  is  made  for  introducing 
"live  steam  at  a  reduced  pressure  between  the  second  and  third  rings,  so 
that  any  leakage  into  the  turbine  will  be  of  steam  and  not  of  air.  The 
labyrinth  packing  between  the  diaphragms  and  hubs  of  the  wheels  is 
shown  in  Fig.  446,  Chap.  XXXI. 

227.  Ridgway  Turbine. — Figure  460  shows  sections  of  the  first  four 
stages  of  a  Ridgway  turbine.  This  shows  the  comparatively  large  blade 
clearances  used  in  these  turbines.  The  clearance  A  is  % Q  in.  or  more; 
B  is  %  m-  °r  more,  and  C  is  J£  in.  or  more. 


CHAPTER  XXXIV 
GENERAL  ARRANGEMENTS  AND  FOUNDATIONS 

Notation. 

h  =  depth  of  foundation  in  feet. 
d  =  diameter  of  foundation  bolts  in  inches. 
Ds  =  diameter    of    standard    steam    engine    cylinder    in    in.    (see 

Chapters  XII  and  XIII). 
Wp  =  weight  of  foundation  in  pounds. 
WE  =  weight  of  engine  in  pounds. 

v  =  volume  of  foundation  in  cubic  feet. 
N  =  revolutions  per  minute. 

228.  General  Arrangements. — For  ordinary  standard  engines  general 
arrangement  drawings  are  not  usually  required.     For  special  work  or  for 
some  special  arrangement  of  piping  they  are  required.     In  any  case, 
especially  with  large  engines  of  the  cross-compound,  twin  or  duplex  types, 
such  drawings  are  a  useful  check  and  are  valuable  for  the  erecting  engineer. 

A  simple  layout  of  a  Bass-Corliss  engine  is  shown  in  Fig.  461.  The 
engine  is  a  26  and  54  by  48  in.  cross-compound,  provided  with  a  rope  wheel. 
The  wheel  is  in  four  parts,  each  having  its  own  hub  and  arms  and  bolted 
together  at  the  flanges  Two  wheels  carry  eight  2  in.  ropes  each  and  the 
others  each  carry  nine  1J^  in.  ropes. 

The  receiver  and  piping  are  under  the  floor,  and  so  arranged  that  the 
connecting  rod  of  either  engine  may  be  disconnected  and  the  other  engine 
continued  in  operation.  Should  the  low-pressure  engine  carry  the  load, 
live  steam  which  has  been  passed  through  a  reducing  valve  is  admitted  to 
the  receiver.  Engines  so  built  by  the  Bass  Foundry  and  Machine  Co. 
are  heavier  than  the  ordinary  compound  engine;  the  long-range  cut-off 
gear  is  used  so  that  one  engine  may  carry  the  normal  load  continuously 
with  a  reasonable  overload  in  case  of  accident  to  the  other  side.  This  is 
desirable  in  plants  with  few  engines — perhaps  only  one — in  which  an  acci- 
dent requiring  a  complete  shut-down  would  be  serious. 

229.  Foundations  are  provided  to  secure  the  engine  and  to  absorb 
vibration.     The  mass  required  for  the  latter  depends  upon  a  number  of 
factors,  but  aside  from  cost,  it  is  better  to  have  too  great  than  too  small 

679 


680 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


a  mass.  In  fact,  greater  cost  in  building  the  foundation  may  sometimes 
prevent  far  greater  cost  in  corrections  made  necessary  by  excessive 
vibration. 

A  formula  for  extreme  foundation  depth  devised  by  the  author  some 
years  ago  for  Corliss  engines  is : 

(1) 


FIG.  461. — General  arrangement  of  Bass-Corliss  engine. 

where  h  is  the  depth  in  feet  and  Ds  the  diameter  of  the  standard  cylinder 
used  in  Chaps.  XII  and  XIII.  This  formula  is  entirely  empirical  and  in- 
tended for  ordinary  conditions  of  soil,  etc.  It  has  been  used  for  many 
large  Corliss  engines  and  so  far  as  is  known  has  proved  satisfactory. 
This  may  be  due  to  the  fact  that  the  proportions  are  rather  massive,  but 
concrete  is  comparatively  cheap. 


GENERAL  ARRANGEMENTS  AND  FOUNDATIONS 


681 


A  formula  of  a  more  scientific  form  is  given  by  E.  W.  Roberts  in  his 
Gas  Engine  Handbook,  which,  with  change  of  notation  is: 


WF  =  KW 


(2) 


WF  is  the  required  weight  of  foundation  in  lb.,  WE  the  weight  of  the  engine 
and  N  the  r.p.m.  The  value  of  K  in  the  seventh  edition  of  the  handbook 
is  0.35. 

After  consulting  with  ten  leading  builders  of  stationary  gas  engines  in 
this  country,  Mr.  Roberts  calculated  values  of  K  from  their  data  for 
various  types  of  engines.  These  values,  with  much  other  valuable 
matter  relating  to  foundations  was  published  in  The  Gas  Engine  for 
September,  1916.  The  volume  in  cu.  ft.  of  a  concrete  foundation  is: 


v  =  kW 


(3) 


Mr.  Roberts  does  not  consider  these  investigations  final,  but  at  present 
they  are  the  best  effort  that  has  been  made  toward  the  solution  of  the  in- 
ternal-combustion-engine foundation  problem.  The  values  of  K  and  k 
proposed  by  him  are  given  in  Table  103. 


TABLE  103 


Type  of  engine 

K 

k 

4-cylinder  vertical  gas  engine  

0.130 

0  .  000975 

3-cylinder  vertical  gas  engine.        .      .            

0  150 

0.001130 

2-cylinder  vertical  gas  engine 

0  175 

0  001310 

4-cylinder  vertical  Diesel  engine 

0  177 

0  001330 

Single-crank,  double-acting  tandem  

0.320 

0.002400 

Twin-crank,  double-acting  tandem            

0.190 

0.001430 

Single-cylinder  horizontal  semi-  Diesel 

0  300 

0  002250 

2-cylinder  horizontal  semi-  Diesel 

0  240 

0  001800 

3-cylinder,  horizontal  semi-  Diesel  
4-cylinder  horizontal  semi-  Diesel     

0.230 
0.225 

0.001730 
0.001690 

2-cycle  horizontal  semi-  Diesel                                      .      .  •  • 

0.230 

0.001730 

Material — Foundations  were  formerly  made  of  brick  but  concrete  is 
now  mostly  used.  A  mixture  of  1  part  of  fresh  Portland  cement,  3  parts 
of  sharp  clean  sand  and  5  parts  of  gravel  or  crushed  stone  is  commonly 
used.  In  some  cases  a  1,  2,  4  mixture  is  used;  this  is  stronger  and  may  be 
better  under  certain  conditions. 

Bolts  and  Washers. — The  determination  of  the  size  of  foundation  bolts 


G82 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


is  arbitrary;  however,  if  too  small  they  break,  causing  a  great  deal  of 
trouble.     For  Corliss  engines  the  author  devised  the  formula: 

,7  ft   -J.    1  tA\ 

d  =  TF  +  o  W 


3*r 
PTI 


where  d  is  the  bolt  diameter  in  inches 
and  Ds  the  diameter  of  the  standard 
cylinder  as  before.  The  number  of  bolts 
properly  influences  the  diameter,  but  in 
(4)  it  is  assumed  that  the  usual  number 
and  arrangement  of  bolts  are  used. 
Square  nuts  are  used  on  the  lower  ends 
of  the  bolts  as  these  have  a  greater  dis- 
tance across  corners,  and  do  not  require 
such  small  clearances  in  the  lugs  of  the 
washer.  Hexagon  nuts  are  used  at  the 
"upper  end  as  they  require  less  space 
and  are  more  easily  handled  with  a 
wrench. 

A  simple  foundation  washer  is  shown 
in    Fig.    462,    for    which    the   following 

formulas  apply;  the  notation  is  given  on  the  figure  and  d  is  the  diameter 

of  the  bolt  as  before. 


o 

FIG.  462. 


=  d 


^+4     B = d 


D  =  l.W+^tol|)  ;  . 

The  dimensions  are  given  in  Table  104.     Sometimes  the  lugs  which 


TABLE  104 


d,  in. 


B,  in. 


C,  in. 


Z),  in. 


E,  in. 


G1,  in. 


IK 


2K 

2% 
3 

3M 

3K 
3% 


9 
LO 
12 
13 
15 
16 
18 
19 
21 
22 
24 


4 

4% 
4% 

5% 


IK 


2 

2K 


4K 


6 

6% 

7% 
8 


IOK 
11 


IK 

2 

2K 
2% 
3 

3K 
4 


K 
1 

IK 


IK 


2 
2K 


GENERAL  ARRANGEMENTS  AND  FOUNDATIONS  683 

hold  the  nut  are  made  deeper  and  are  connected,  forming  a  pocket.  Then 
should  a  bolt  break  at  the  thread  at  the  upper  end  it  may  be  unscrewed 
from  the  lower  nut  and  replaced.  However,  should  it  break  well  down  in 
the  foundation,  the  pocket  would  be  a  detriment  as  will  be  shown 
presently. 

Construction. — Foundation  plans  are  furnished  by  the  engine  builder. 
In  some  instances  drawings  are  furnished  of  wooden  templets  which  are 
located  over  the  foundation  and  hold  the  bolts  in  position  while  the  founda- 
tion is  under  construction.  The  foundation  should  be  below  the  frost 
line  if  exposed,  but  they  are  often  in  a  basement.  In  small  foundations 
the  bolts  and  washers  are  sometimes  embedded  in  the  concrete;  but  it  is 
better  practice  to  have  the  bolt  in  a  pipe,  or  in  a  box  formed  by  four 
boards.  This  box  is  smaller  at  the  bottom  than  at  the  top  and  may  be 
removed  after  the  foundation  is  finished  if  desired.  The  pipe  or  box 
allows  the  bolt  to  be  moved  in  any  direction  to  allow  for  discrepancies 
in  the  engine  bed  holes. 

In  large  foundations  the  washers  are  placed  in  pockets.  Tunnels, 
through  which  a  man  may  crawl,  lead  to  these  pockets.  Such  a  founda- 
tion by  the  Bass  Foundry  and  Machine  Co.  is  shown  in  Fig.  463.  This 
was  built  of  brick  for  a  22  by  42  in.  Corliss  engine  designed  to  run  100 
r.p.m.  The  depth  was  made  according  to  (1),  and  the  foundation  bolts 
of  the  frame  by  (4).  The  boxes  for  the  bolts  are  not  shown,  but  a  note 
on  the  drawing  calls  for  holes  through  brickwork  4  in.  square.  Should 
a  bolt  break  the  lower  end  will  drop  down  in  the  pocket  in  the  brickwork 
if  no  pocket  is  used  on  the  washer.  The  nut  may  then  be  unscrewed  by  a 
man  in  the  tunnel,  and  an  ingenious  mill-wright  or  engineer  will  find  some 
means  of  removing  it.  By  removing  the  washer  a  small  chain  may 
be  let  down  from  the  top  and  fastened  around  the  thread  at  the  lower 
end. 

Grouting. — A  certain  amount  of  space — commonly  1  in. — is  left  be- 
tween the  top  of  the  foundation  and  the  engine  frame  for  grouting.  The 
engine  is  supported  on  wedges  by  means  of  which  it  may  be  raised  or 
lowered  until  it  is  properly  leveled ;  then  the  grouting  is  poured  in.  This 
may  be  one  of  several  substances.  Sulphur  has  been  used  for  small 
engines;  and  for  heavy  mill  engines,  iron  and  steel  turnings  are  rusted 
together  with  sal-ammoniac.  Cement  is  now  the  most  common  sub- 
stance used.  When  this  is  well  set,  the  wedges  are  removed  and  the 
nuts  on  the  foundation  bolts  tightened. 

The  Soil. — Ketchem,  in  his  Structural  Engineers'  Handbook,  gives 
the  following  values  of  maximum  allowable  pressure  on  the  soil  in  tons 
per  sq.  ft. : 


684  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


GENERAL  ARRANGEMENTS  AND  FOUNDATIONS  685 

Ordinary  clay,  and  dry  sand  mixed  with  clay 2 

Dry  sand  and  dry  clay 3 

Hard  clay  and  firm,  close  sand 4 

Firm,  coarse  sand  and  gravel , .  .  5 

Shale  rock 8 

Hard  rock 20 

All  inferior  soils  such  as  loam 1 

While  rock  is  an  ideal  foundation  bed  in  many  ways,  vibration  may  be 
transmitted  through  it  to  the  walls  of  the  building.  It  has  been  advised 
to  make  a  pocket  in  the  rock,  in  which  a  layer  of  sand  is  placed  before 
starting  the  foundation  footings. 

Vibration  may  be  transmitted  through  poor  spongy  soil,  and  if  the 
plant  is  in  a  residence  district  this  may  be  very  objectionable.  Increasing 
the  weight  of  the  foundation  will  help,  but  this  will  not  always  suffice. 
Balancing  the  engine  seems  to  be  the  best  remedy  for  this;  in  fact,  much 
lighter  foundations  may  be  used  with  well-balanced  engines  in  any  case. 

Horizontal  vibration  is  usually  the  most  troublesome.  In  a  horizontal 
engine  this  may  be  remedied  largely  by  balancing  all  of  the  reciprocating 
parts.  In  a  vertical  engine  the  revolving  parts  only  should  be  balanced ; 
but  there  remains  the  vibration  due  to  the  turning  effort  which  can  not  be 
balanced. 

It  has  usually  been  considered  bad  practice  to  build  an  engine  founda- 
tion in  contact  with  the  walls  of  the  building;  but  Roberts,  in  the  article 
referred  to,  tells  of  a  consulting  engineer  in  Chicago  who  does  this,  his 
idea  being  to  increase  the  mass  that  must  be  set  in  motion  by  the  unbal- 
anced forces  of  the  engine.  He  further  claimed  that  the  results  were 
entirely  satisfactory. 

The  opposite  practice  of  isolating  the  foundation  by  a  bed  of  sand  has 
already  been  referred  to.  Roberts  tells  of  the  use  of  cork  for  this  pur- 
pose, and  a  cork  sheeting  is  being  used  in  this  country.  A  concrete  pit 
with  a  heavy  bottom  is  first  made;  the  sides  and  walls  are  lined  with  this 
sheet  cork;  then  into  this  the  foundation  proper  is  poured. 

The  following  foundation  specification  was  furnished  by  the  Buckeye 
Engine  Co.  for  a  15J4  by  18  in.  vertical  steam  engine.  The  depth  is  4  ft., 
and  about  15  in.  margin  is  left  around  the  bed  casting.  There  are  eight 
1%  in.  bolts  and  the  allowance  for  grouting  is  1  in. 

SPECIFICATION  FOR  FOUNDATION 

Build  the  foundation  of  a  good  concrete  mixture  as  follows:  5  parts 
clean  gravel  or  broken  stone;  3  parts  clean  sharp  sand,  and  1  part  of 
good  grade  Portland  cement.  Mix  thoroughly  and  ram  in.  In  building, 


686  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

leave  room  around  each  bolt  hole  so  that  bolts  can  be  moved  ^  in.  in  any 
direction.  This  is  best  done  by  placing  a  pipe  or  a  wooden  box  around 
bolts,  which  can  be  taken  out  after  foundation  is  finished,  or  lifted  up  as 
the  building  of  the  foundation  progresses.  The  drawings  represent  a 
sufficient  amount  of  masonry  to  give  the  proper  weight,  but  it  must  be  on 
solid  ground.  If  sufficiently  solid  ground  is  not  obtained  at  the  depth 
shown  by  drawing,  the  excavation  must  be  continued  deeper;  the  width 
must  also  be  increased.  The  additional  depth  is  to  be  filled  with  the 
same  concrete  mixture.  In  many  cases  however,  sufficient  solidity  may 
be  had  by  a  thorough  ramming  of  the  ground. 

The  copings  can  be  made  of  the  same  material  or  cut  stone,  and  the 
whole  should  have  at  least  two  weeks  to  set  before  engine  is  placed. 
Observe  closely  the  drawing  of  "templet"  for  placing  foundation  bolts. 

References 

American  foundation  practice   The  Gas  Engine,  Sept.,  1916. 

Exhaust  vibration TheGas  Engine,  Oct.,  1916. 


CHAPTER  XXXV 
DESIGN  METHODS 

230.  Personal  efficiency  is  one  of  the  first  requisites  of  the  designing 
engineer.  Many  men  are  able  to  promote  the  business  of  others  to  per- 
fection but  are  exceedingly  negligent  of  their  own.  A  designer  should 
have  a  well-organized  system  of  his  own  whether  the  concern  by  whom  he 
is  employed  has  such  a  system  or  not.  He  must  be  orderly,  consistent 
and  thorough,  and  must  not  take  things  for  granted.  His  own  records 
should  be  well  kept  as  well  as  those  he  keeps  for  the  company.  He  should 
have  note  and  data  books  for  different  phases  of  his  work.  He  should 
have  the  best  handbooks  and  design  manuals  and  be  familiar  with  them. 
His  working  formulas  should  be  well  selected. 

His  drafting  instruments  should  be  kept  in  good  condition,  and  al- 
though he  attains  to  a  position  in  which  he  is  required  to  do  little  or  no 
drafting,  he  should  never  feel  above  working  over  the  board  occasionally, 
for  unless  he  has  traversed  this  path  he  is  hardly  fit  to  direct  the  efforts  of 
those  who  do  this  kind  of  work. 

The  designing  engineer  should  never  get  beyond  study.  He  should 
keep  abreast  of  the  times  in  his  special  work  and  should  strive  to  broaden 
out  some.  Technical  periodicals  help  in  this,  as  does  also  membership  in 
an  engineering  society — notably  one  of  the  national  societies  such  as  the 
American  Society  of  Mechanical  Engineers.  He  should  be  ready  to  lead 
in  thought  as  far  as  he  is  permitted  to  do  so  and  must  not  simply  drift 
along  in  the  regular  routine  of  the  office. 

He  should  use  a  good  slide  rule  such  as  the  Log  Log  Duplex;  he  can 
then  do  more  work  with  less  fatigue  and  therefore  of  improved  quality. 

Some  firms  require  important  calculations  to  be  kept  in  note  books 
which  are  the  property  of  the  company.  This  is  good  practice,  and  if 
required  should  be  attended  to  conscientiously.  If  not  required  it  is 
well  for  the  designer  to  keep  such  data  himself. 

If  in  charge  of  the  department  and  responsible  for  shop  orders,  parts 
most  needed  or  requiring  the  most  time  to  obtain  should  be  ordered  first ; 
this  includes  large  or  difficult  castings,  large  forgings,  and  sometimes  steel 
castings.  It  is  sometimes  difficult  to  do  this  as  some  of  these  parts 
are  logically  the  last  to  be  designed,  but  with  proper  system  in  designing 
this  can  usually  be  managed,  so  that  the  work  may  be  started  before  the 

687 


688 


DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 


design  is  complete.  This  is  not  the  most  desirable  practice  from  the  de- 
signer's standpoint,  but  the  writer  is  sure  that  it  may  often  be  done  to 
great  advantage. 

Printed  lists  are  usually  furnished  for  the  parts  built  by  different 
departments.  If  this  is  not  done,  a  list  may  easily  be  made  on  tracing 
cloth  with  spots  blocked  for  drawing  or  sketch  numbers.  In  this  way 
nothing  will  be  overlooked.  A  list  of  Corliss  engine  parts  prepared  for  the 
forge  shop  is  given  in  Table  105.  Space  is  left  for  the  machine  shop  draw- 
ing showing  the  finished  part  as  a  matter  of  convenience  to  the  office. 

TABLE  105. — THE  BASS  FOUNDRY  AND  MACHINE  COMPANY,  ENGINEERING 

DEPARTMENT 

Subject  FORGINGS  Order 

Answering  SMITH-SHOP  Fort  Wayne,  Ind 


Name  of  part 


Drawing 

No. 


Sketch 
No. 


Pieces 


Crank  shaft 

Connecting  rod 

Eccentric  rod 

Reach  rod 

Piston  rod 

Valve  stems 

Crank  pin 

Crosshead  pin 

Crosshead  key 

Hook  bolts,  main  bearing 

T-head  bolts,  main  bearing 

Hook  bolts,  outer  bearing 

T-head  bolts,  outer  bearing 

Extension  bed  links 

Girder  bolts 

Flywheel  hub  bolts 

Flywheel  rim  bolts 

Flywheel  links  bolts 

Flywheel  keys 

Rope  wheel  hub  bolts 

Rope  wheel  rim  bolts 

Rope  wheel  tie  rods 

Rope  wheel  keys 

Gear  bolts 

Gear  links 

Gear  keys 

Carrier  shaft 

Foundation  bolts 

Eccentric  strap  bolts 

Conn,  rod  wedge  bolts 


DESIGN  METHODS 


689 


The  smith  shop  sketch  numbers,  number  of  pieces  and  the  date  of  order 
are  also  given.  A  duplicate  copy  should  be  kept  in  the  office,  and  when  a 
piece  is  ordered,  this  may  be  signed  by  the  smith  shop  clerk  when  the 
sketch  is  delivered.  Similar  lists  may  be  made  for  other  departments. 

Standards. — Much  work  in  heat  engine  design  may  be  standardized, 
and  this  has  been  shown  through  the  chapters  on  machine  design.  The 
application  of  standard  tables  to  other  pressures  and  to  compound  engines 
is  explained  in  Chaps.  XII  and  XIII.  The  main  dimensions  may  be  in- 


TABLE  106. — STANDARD  ENGINES 


Name  of  Part 


Drawing  No. 


Cylinder 

Girder 

Guide  barrel — No. 

Main  bearing — No. 

Outer  bearing 

Cylinder  heads 

Cylinder  foot 

Piston 

Crosshead 

Crank  . 


Size  of  engine 

Class  of  engine 

Main  bearing — diam ,  length. 


~ 
Crosshead 


{  Screw  &  Locknut 

I  Taper  fit  &  key 


(Arm 
(Disk 


Wrist  plate 


Carrier 

Center  foot 


Cent,  of  cyl.  to  cent,  of  shaft  = 

Cent,  of  eng.  to  bottom  of  castings 

Remarks : 


I  Double 
{  Single 
t  Double 


Eccentric   

Valves 
Bonnets 
Valve  gear 
Dashpots 
Release  rig 
Carrier  pin  stub 
Valve  gear  pin  stub 
Governor 

Connecting  rod. .  .  . 

Eccen.  &  reach  rods 

Piston  rod 

Valve  diagram  .... 


Si 
1  Double 


(Solid 
}gtrap 


f  Single 
]_   °,  . 
Double 


690  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

eluded  in  one  table  for  engines  designed  for  a  standard  pressure,  making 
it  convenient  in  furnishing  data  for  specifications.  Horsepower  tables 
may  also  be  computed,  but  such  calculations  are  so  quickly  made  with  a 
slide  rule  that  this  may  be  of  little  use. 

As  soon  as  an  engine  of  any  size  has  been  standardized  it  is  of  great 
convenience  to  enter  the  drawing  numbers  in  a  blank  similar  to  Table  106, 
which  was  prepared  for  Corliss  engines.  This  greatly  facilitates  getting 
orders  into  the  shop. 

In  furnishing  date  to  salesmen  for  work  not  included  in  standard 
tables,  the  blank  shown  in  Table  107  has  been  found  useful. 

TABLE  107 

Subject  Order 

Answering 

Fort  Wayne,  Ind. 
Maximum .  . 


Regular Piston  speed 


Steam  pressure Ibs.  R.p.m. 

Minimum 

Economical  horsepower  with cut-off  in  h.p.  cyl 

Maximum  horsepower  with cut-off  in  h.p.  cyl 

Run  over  or  under Belt  forward  or  back 

Diam.  of  wheel Width  of  face  for  belt 

No.  of  ropes Diam Weight  of  wheel Ibs. 

Main  bearing Outer  bearing Counter  shaft  bearing 

Main  shaft  diam swelled  to 

Count,  shaft  diam swelled  to 

Guide  barrel Bed 

Diam.  steam  inlet  to  cyl.  h.p l.p ~         ,    .  .       f  H.p 

~  A,  .  ,  ,  Face  of  piston  i  T 

Diam.  steam  outlet  to  cyl.  h.p l.p I  L.p 

,  J  H.p ~       ,      ,     .    1  Diam „      ,          {  Diam 

Diam.  piston  rod  i  T  Crosshead  pin  j  T        ,,  Crank  pin  \  ,       A. 

(  L.p I  Length I  Length 

Connecting  rod  diam.  at  neck At  center 

Arranged  so  that  h.p.  or  l.p.  engine  may  run  alone  and  carry  whole  load 

Further  data : 

Partial-assembly  Sketches. — Sketches  of  groups  of  parts  are  of  great 
convenience  in  design.  They  are  also  helpful  in  checking  existing  design, 
as  any  interference  or  other  discrepancy  is  readily  seen.  These  may  be 
made  on  tracing  cloth  and  a  set  of  blueprints  blocked  for  inserting  di- 
mensions used  for  each  case.  These  may  be  kept  on  file  for  reference. 
If  certain  dimensions  are  not  required  a  dash  may  be  used.  These 
sketches,  with  the  standard  tables  previously  referred  to  are  useful  in 
getting  out  large  castings — such  as  the  frame — much  sooner  than  can  be 
done  by  the  ordinary  process  of  design.  Figs.  464  and  465  are  part  of  a 


DESIGN  METHODS 


691 


FIG.  464. 


Fio.  465. 


692  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

set  of  such  sketches  used  by  the  author  in  Corliss  engine  design.  They 
are  self-explanatory. 

The  above  suggestions  are  given  as  a  few  examples  of  how  work  may 
not  only  be  expedited,  but  the  liability  of  error  lessened.  A  good  system 
will  be  found  in  a  large  number  of  places,  but  if  not,  the  suggestions  may 
be  helpful.  The  lists  given  are  more  applicable  to  smaller  plants,  but 
the  same  principle  will  apply  to  plants  of  any  size. 

If  data  is  looked  up  for  a  given  problem,  it  is  well  to  put  it  in  a  data 
book  in  good  form  for  future  reference.  The  same  may  be  said  of 
formulas  derived,  care  being  taken  to  give  all  notation  in  the  proper 
units.  They  are  then  readily  available  should  such  problems  again 
arise. 

Many  other  ways  will  suggest  themselves  if  one  starts  to  systemize, 
but  all  system  should  be  as  simple  as  possible  so  that  it  may  not  become 
a  burden. 


APPENDIX  1 
MOMENT  OF  INERTIA  OF  IRREGULAR  SECTIONS 

A  graphical  method  of  finding  the  moment  of  inertia  is  given  in  Mor- 
ley's  "  Strength  of  Materials, "  with  a  full  explanation  of  the  theory.  The 
method  will  be  briefly  described  here  without  proof. 

In  Fig.  466,  the  outside  irregular  line  bounds  the  section  of  area  A, 
whose  moment  of  inertia  is  required.  Draw  any  axis  XX  below  the 
figure  normal  to  the  plane  of  bending  (parallel  to  the  neutral  axis); 
also  draw  line  SS  any  distance  above  the  figure.  Across  the  figure  at 
any  point  draw  line  PQ  parallel  to  XX.  Where  line  PQ  cuts  the  outline 


N  N, 


M/M 


FIG.  466. 

of  area  A,  erect  perpendiculars  PN  and  QM  as  shown.  Connect  points 
N  and  M  to  any  point  0  in  line  XX.  Lines  NO  and  MO  cut  line  PQ 
at  PI  and  Qi,  and  these  are  points  in  an  interior  area  AI.  By  drawing 
a  number  of  lines  PQ,  enough  points  may  be  found  through  which  to 
draw  the  curve  enclosing  area  A\. 

By  erecting  perpendiculars  from  PI  and  Qi  and  connecting  points  NI 
and  MI  with  0,  points  P2  and  (?2  are  found,  locating  points  in  a  third 
area  A2. 

Having  found  the  three  areas,  and  knowing  the  distance  d  between 
lines  SS  and  XX,  we  may  proceed  as  follows:  The  distance  of  the 
center  of  gravity  of  the  section  (of  area  A)  from  line  XX  is: 


(D 


693 


694  DESIGN  AND  CONSTRUCTION  OF  HEAT  ENGINES 

The  moment  of  inertia  about  axis  XX  is: 

Ix  =  Azd*  (2) 

And  about  neutral  axis  GG: 

IQ  =  Ix  -  Ay*  (3) 

The  modulus  of  section  is: 

.         z=^  ^:v:         (4) 

where  c  may  be  either  cs  or  cx ;  if  the  elastic  limit  in  tension  and  com- 
pression is  the  same,  the  larger  value  of  c  must  be  used.  If  M  is  the  bend- 
ing moment  in  Ib.  in.,  and  S  the  stress  in  Ib.  per  sq.  in.: 

M  =  *±  (5) 

C 

and 

«       Me  ,  . 

b  =  -j-  (6) 

JG 

The  areas  may  be  found  by  a  planimeter.  If  not  drawn  actual  size 
and  one  inch  on  the  drawing  represents  n  actual  inches,  d  must  be  multi- 
plied by  n,  and  all  areas  found  by  the  planimeter  by  n2;  then  these 
values  may  be  used  in  the  formulas. 


APPENDIX 


695 


I 

I 
1 1 

Z,  § 
0Q 

5  s 

*  2 

s-. 

«•! 

^      o  V 
w      « ^ 

5    11 

|z      5-5 

I    *(5 

*  | 

£ 

§1 


c^ 


&13 

QM4J 


8.2 


cox  t-K  i-N  \co  w\  \oo  \IM  i 

N        O*       CO       C0\   rH        «\   r-K 


•^  <*  N 

•O  \<;o     -rj-  \w 

vrH   COX   \5O   rH\ 


>      <N  Tj«               N 

\SO  \tO  \!0  \CO  \C<3  Tj  \to  \w 

Oi\  t-\  \;o  0>\   \00  rH\ 

N  rH  «\    CO       rH\  jvj 


'  >*  ^e   «  \«         Tj<    to  \« 


\ec  VH  \M          \«     to 
Oi\  ^-N  »n\  \oc  i-N  v-i  \T)< 

rH       rH       Cq       t-\   «       rH\   r-l\ 


to 
H 

O>\ 


r>«1ri- 

CO'— ii—i-«^iOiOiOO5COO<M«Oi— lO 

r-^Tr- (OOOOOC<lCOOCDC^COi— IrH 


0  c<  t>  c<  o      r- 


t^C^^i— itOrHi— (tOI>(NOi 

<xT  cT  i-T  wf  cxT  co^  i>T  cf  i>"  co"  oT 


o  o  o  o  o  o  o  o  o  o  o  o  *-!  I-H  TH  T-I 


co  co  -^  »o  co  i^  0    as 


O   O   O   O   O  O   O   o   O  O  O 


(N   <N  C<    (N  CO  CO  CO 


to  <N      <N      <N  N      N      tO 

\rH   V*   \CO   \W   \CO       tO    \N    \00  \«   \CO   \rH  «D       N       « 

W\   rH\   O>\   rH\   CO\  \rH   r-K   1QK  CO\    l-K  «K  VH   \«O   \CO    \00   \N   NflO  «K   rH\  VH  VH   \^0  V£0  V»  VH  Nj-l 

rH       iH       t-\                         C4       C4       rH  i-Hv   «\   O5\   COK   rH\   IO\   N       CO       COK   t^\  «5\  t^\  rH\   UJ\  OS\ 


Nrf<  VH  \oo  VH  \e*  VH  \PO  v*  \oo          \oo  V*  \oo  \^«  \oo  VJ(  \oo 

r-K  "5\  COK  l-K  i-K  0>\  >0\  COK  t-K  r-K  i-K  COK  rH\  >CK  COK  bX 


INDEX 


Absolute  pressure,  10 
Acceleration,  forces  due  to,  296 

of  reciprocating  parts,  297 

Klein's  method,  298 
Adiabatic  expansion  of  gas,  56,  59 

of  steam,  67 
Admission,  8 
Alloy  steel — see  steel. 
Aluminum,  474 
Angular  acceleration  of  connecting  rod, 

299 

Angular  advance,  411 
Angularity  of  connecting  rod,  415 
Area  of  ports,  402 
Automatic  cut-off  engine,  17,  139 
Axial  thrust,  225 


Bearings,  proportions,  115 

steam  turbine,  117 

thrust,  119 

Belt  forward  or  back,  15 
Bending  and  twisting,  combined,  481 
Bilgram  diagram,  417 
Blade  efficiency,  257 
Blades  of  steam  turbines,  24,  216,  217, 

218,  222,  255,  271,  277,  657 
Blading,  impulse — practical  notes  on,  228 
Blast-furnace  gas,  38 
Bleeder  turbine,  30 
Bolts  and  nuts,  U.  S.  Standard,  695 
Bolts,  strength  of,  695 
Brake  horsepower,  53,  98,  187 
Bruce- Macbeth  gas  engine,  43,  190 
Busch-Sulzer  Bros.  Diesel  engine,  45 


Back  pressure,  11,  131 

Balancing,  334 

angle  engine,  341 
connecting  rod,  343 
four-crank  engine,  340 
multi-cylinder  engine,  338 
primary,  334 
reciprocating  parts,  336 
secondary,  337 
six-crank  engine,  342 
three-crank  engine,  340,  341 
turning  effort,  345 
two-crank  engine,  340 

Balance  weights,  346 

Basis  of  design,  493 

Bearing  cap  bolts,  619 

Bearing  wedge  bolts,  617 

Bearings,  clearance,  113 
eccentric,  120 
main,  617 
metal,  114 
oil  grooves,  113 


Cam  valve  gears,  448 

Capacity  curves  of  Continental  engines, 

49 

Cap  bolts,  619 
Card  factor,  135 
Carnot's  cycle,  54 
Cast  iron,  469 
Center  of  gravity,  285 

of  irregular  section,  693 
Classification      of     internal-combustion 
engines,  32 

of  steam  engines,  13 

of  steam  turbines,  24 
Clearance,  79,  131,  202,  503 

in  bearings,  113 

in  internal-combustion  engines,  202 
Collars  for  shafts,  609 
Column  formulas,  485 
Combined  bending  and  twisting,  481 
Combined  indicator  and  inertia  diagrams, 

301,  303 
Combustion,  temperature  rise  due  to,  61 


697 


698 


INDEX 


Compound  steam  engines,  87,  152,   154 
condensing,  155 
cross-compound,  20 
tandem,  20 

Compound  turbines,  30 
Compression,  11,  79,  131,  145,  194,  196, 

304 
effect  upon  capacity  of  steam  engine, 

145 
effect  upon  economy  of  steam  engine, 

79,  145 
effect  upon  smoothness  of  operation, 

304 

in  internal-combustion  engines,  194 
Compression  pressure,  steam  engine,  10, 

131,  135 

internal-combustion  engine,  194 
Condensation,  cylinder,  81 
and  re-evaporation,  78 
affected  by  compounding,  87 
cylinder  ratio,  88 
design  of  cylinder,  82 
range  of  pressure  and  temperature, 

85 

ratio  of  expansion,  85 
size  of  cylinder,  82 
speed,  81 
steam  jacket,  86 
superheated  steam,  90 
Condenser,  265 
Condensing  engine,  155 
Connecting  rod,  285,  343,  544 
body,  545 

center  of  percussion,  286 
ends,  552 

moment  of  inertia  of,  287 
properties  of,  285 
radius  of  gyration  of,  285 
wedge  bolts,  555 

Conservation  of  residual  velocity,  248 
Constant-pressure  cycle,  33,  58 
Constant-volume  cycle,  33,  55 
Continental  motors,  46 
Conventional  indicator  diagrams,   com- 
pound engine,  153 
internal-combustion  engine,  63,  201 
simple  steam  engine,  130 
Cooling  of  cylinder,  37,  41,  94 
Cooling  water,  94 


Corliss  long-range  cut-off,  432 
Corliss  valve  gear,  426 
Counterbore,  503 
Cranks,  572 
Crank  arm,  579 

effort,  289 

hub,  582 

pin,  573,  589 
Crankshaft,  586 
Crosshead,  562 

pin,  563 

shoe,  565 

Critical  speed  of  turbine  shaft,  666 
Curve  of  fuel  consumption,  43,  45,  46 

of  steam  consumption,  22,  86 
Cushion  steam,  133 
Cut-off,  commercial,  129 

long  range,  432 

to  find  from  m.e.p.,  133 
Cycle,  Carnot's,  54 

of  internal-combustion   engine,    32, 
54,  55,  58 

Rankine's,  complete  expansion,  67 
incomplete  expansion,  68 

of  steam  engine,  3,  6 

of  steam  plant,  3 
Cylinder,  499 

condensation  in,  78,  81 

diameter,  compound  engine,  178 
internal-combustion   engine,    187, 

189 
simple  steam  engine,  143 

efficiency,   internal-combustion    en- 
gine, 93 
steam  engine,  78 

feed,  133,  157 

heads,  504 

internal-combustion  engine,  509 

lagging,  507 

lubrication,  122 

ratio  in  compound  engines,  88,  154, 
155,  157 

steam  engine,  508 

walls,  strength  of,  478,  499 


D 


Dashpots,  375,  459 
Dead  center,  8 


INDEX 


DeLaVergne  oil  engine,  43 

Design  methods,  687 

Diagram  efficiency,  225,  226   256 

Diagram  factor,  33,  34,  40,  58,  192 

Dimensions   and    data  for    Continental 

engines,  46,  49 

for  Bruce- Macbeth  gas  engine,  44 
Double-ported  valves,  419 
Dry  saturated  steam,  66 


E 


Eccentrics,  408 

shifting  and  swinging,  420 
Economy,  70 

comparative,  76 

internal-combustion  engine,  75 

prime  mover,  71 

steam  engine,  71 

steam  plant,  70 

steam  turbine,  71 
Efficiency,  brake,  185 

cylinder,  78 

Diesel  cycle,  59 

economic,  185 

hydraulic,  237 

mechanical,  54,  97,  185,  191,  192 

Otto  cycle,  56 

Rankine  cycle,  67,  68 

ratio  of  internal-combustion  engine, 
76 

thermal,  56,  59 

volumetric,  60,  189 
Electrical  horsepower,  53,  209 
Engine  classification,  13,  32 
Entropy  diagram,  57,  60,  236,  237,  245 
Entropy  of  steam,  67 
Equivalent  eccentric,  440,  444 
Exhaust  muffler,  41 
Expansion  of  steam,  9 


Flywheel,  350,  623 

arms,  bending  of,  628,  629 
virtual  number,  629 

constants,  354 

hubs,  636 

hub  bolts,  637 

hub  shear  bolts,  638 

methods  of  construction,  640 

rim  bolts  and  links,  639 

rim  and  arm  sections,  633 

strength  of,  623 

Unwin's  formulas  for,  626 

weight,  comparison  of  methods,  363 
displacement  method,  356 
velocity  method,  351 
Formation  of  steam  under  constant  pres- 
sure, 65 

Formulas,  selection  of,  475 
Forced  fits,  537 
Foundation,  679 

bolts  and  washers,  681 

construction,  683 

depth,  680 

grouting,  683 

material,  681 

specification,  685 

weight  and  volume,  681 
Four-cycle  engine,  35 
Frame,  611 

center-crank,  611 

reactions  on,  321 

side-crank,  613 

stresses  in,  611 
Friction,  104 

effect  on  flow  of  steam,  68,  213 

laws   for   dry   and   lubricated   sur- 
faces, 104 

Friction  horsepower,  53,  98 
Fuel  consumption,  43,  45,  46,  193 
Fuel  nozzle,  456 


Factor  of  judgment,  486 
Factor  of  safety,  464,  493,  497 
Filtration  of  lubricating  oil,  122 
Flat  plate  formulas,  501 
Flow  of  steam,  68 


G 


Gas  consumption — see  fuel  consumption 
Gas  engine,  33,  38 

cycles,  54 

Gas  velocity,  95,  407 
General  arrangements,  679 


700 


INDEX 


Governing,  compound  engines,  172 

Diesel  engines,  454 

hit-and-miss,  40,  196 

internal-combustion  engines,  40,  196, 
400,  454 

quality,  41,  196 

quantity,  41,  197 

steam  engines,  16,  138 

steam  turbines,  31,  267 
Governor,  367 

centrifugal,  369,  377,  378,  390,  395 

conical,  371,  378 

dashpots,  375 

examples  from  practice,  387 

formulas,  centrifugal  governors,  377 
conical  governors,  378 
speed  fluctuation,  375 
speed  variation,  376 
spring  governors,  382 

gravity  balance,  387 

hunting,  386 

inertia,  371,  395 

isochronous,  370 

machine  design,  387 

practical  considerations,  386 

relay,  374,  396 

safety,  374,  400 

selection  of  constants  of  regulation, 
387 

speed  adjustment,  375 

spring,  371,  382,  387,  390,  395 

stable,  370 
Grease,  109 

cups,  112 
Guide,  pressure  on,  300 


H 


Heat  engine,  commercially  successful,  1 

important,  1 
Heat  factor,  73,  209,  237 

per  stage,  237 
Heating  value  of  fuel,  188,  196 

higher  and  lower,  196 
Heat  of  the  liquid,  66 
Heat  supplied  in  formation  of  steam,  66 
Heavy-oil  engine,  33,  39 
High  steam  pressure,  91 
Hit-and-miss  governing,  40,  196,  354 


Horsepower,  brake,  53,  187,  189 

compound  engine,  178 

electrical,  53,  209 

friction,  53 

limit  of,  144,  193 

indicated,  52,  141,  178,  187 

internal-combustion  engine,  186 
simple  formulas  for,  189 

locomotive,  144 

simple  steam  engine,  141 

steam  turbine,  208 
Hoop  stress,  625 
Hot  bulb  engine,  39 
Hot  plate  engine,  39 
Hubs,  582,  636 
Hub  bolts,  637,  638 
Hydraulic  efficiency,  237 


Ignition,  34,  453 
Impulse,  23 

turbine,  24,  26,  222,  268 
Indicated  horsepower,  52,  98,  141,  178, 
186 

compound  engine,  178 

internal-combustion  engine,  186 

simple  steam  engine,  141 
Indicator  diagrams,  51 

compound  engine,  153-173 

internal-combustion      engine,      34, 
43-46,  55-60,  197-204 

simple  steam  engine,  10-21,  128 
Inertia,  296 

diagrams,  297 

and  indicator  diagrams  combined, 

301 

Initial  condensation,  78 
Initial  pressure,  10 
Inlet  and  exhaust  passages,  503 
Intake  muffler,  42 
Internal-combustion  engine,  32,  185 

2-cylinder,  319 

3-cylinder,  320 

4-cylinder,  321 

6-cylinder,  322 

8-cylinder,  323 

12-cylinder,  323 

2-cycle,  35,  37,  453 


INDEX 


701 


Internal-combustion  engine,  4-cycle,  33 

cycles,  32,  54 

data,  43,  191 

Diesel,  34 

gas,  38 

heavy-oil,  39 

hot  bulb,  or  plate,  39 

light-oil,  38 

starting,  41 
Irregular  section,  center  of  gravity  of,  693 

moment  of  inertia  of,  693 

modulus  of  section  of,  693 


Jacket  water,  37,  94 

practical  suggestions,  95 


Keys,  for  cranks,  583 
for  flywheels,  640 
Kilowatts,  209 


Lagging,  cylinder,  507 

Lap,  411 

Latent  heat  of  saturated  steam,  67 

Lead,  8,  411 

in  steam  turbines,  250 

variable,  445 
Light-oil  engines,  33,  38 
Limits  of  horsepower,  144,  193 
Lubricants,  107 

acid  test,  108 

cold  test,  108 

density,  108 

fire  test,  108 

flash  point,  108 

friction  test,  108 

grease,  109- 

gumming  test,  108 

viscosity,  108 
Lubricating  systems,  109 

forced  feed  or  continuous,  109,  121 

individual,  109 

intermittent  feed,  109 

restricted  feed,  109 

splash  system,  110 


Lubrication,  cylinder,  123 
forced,  99 
syphon,  99 
and  pad,  99 

M 

Machine  design,  463 
Materials,  properties  of,  470 
Maximum  horsepower,  144,  193 
Maximum  strain  theory,  477 
Mclntosh  and   Seymour   Diesel   engine, 

45 
Mean  effective  pressure,  52 

compound  engine,  155,  158 

gas  and  oil  engines,  63, 186,  192,  190 

Rankine  cycle,  67 

simple  steam  engine,  130 
Mechanical  efficiency,  54,  97,  191,  192 
Mechanics,  283 

of  slider  crank,  283 
Mechanism  of  steam  engine,  5 
Methods  of  design,  687 
Modulus  of  section  of  irregular  section, 

694 
Moment  of  inertia  of  connecting  rod,  286 

of  irregular  section,  693 
Mufflers,  41,  42 


N 


Noncondensing  steam  engine,  20 
Nozzles,  fuel,  456 

and  other  passages,  210 
practical  notes  on,  219 

steam  turbine,  210,  219,  274-276 

O 

Operation  of  steam  engine,  6 
Oil  grooves,  113 
Oil  guards,  112 
Otto  cycle,  33 


Peabody's  direct  method,  238 
Piston,  515 

acceleration  of,  296,  298 


702 


INDEX 


Piston,  box,  516,  525,  526 

built-up,  524 

conical,  517,  526 

internal-combustion  engine,  529 

rings,  518 

rod,  530 

formulas,  531 

speed,  82,  83,  95,  143,  144,  191 

thrust,  compound  engine,  175 
internal-combustion  engine,  201 
simple  steam  engine,  140 

trunk,  527,  528 

velocity  of,  287 
Poisson's  ratio,  476 
Port  area,  402 
Power  of  heat  engines,  51 

internal-combustion  engine,  49,  186 

steam  engine,  208 

steam  turbine,  208 
Power  plant,  3 
Practical  cycles  using  gas,  54 
Pressed  fits,  537 
Pressure  and  temperature  of  saturated 

steam,  66 
Producer  gas,  38 
Properties  of  materials,  470 


Q 


Quality  governing,  41,  196 
Quantity  governing,  41,  197 


Radius  of  gyration,  285,  286 
Rankine  cycle  for  complete  expansion, 
67 

for  incomplete  expansion,  68 
Rating  of  internal-combustion  engines, 
198 

of  steam  turbines,  267 
Ratio  of  expansion,  10,  85 
Ratio  of  stroke  to  cylinder  diameter,  82, 

95,  143,  192 
Reaction,  23 
Reaction  turbine,  27,  254,  279 

practical  notes  on,  260 
Receiver,  19 

influence  of,  159 


Reciprocating   engine — passing  not   im- 
minent, 2 

Rectangular  section  in  torsion,  484 
Rectangular   valve   diagram,    422,    425, 

457 

Re-evaporation,  78 
Reheat  factor,  237 
Release,  11 

Releasing  gear,  434,  453 
Reversal  of  thrust,  304 
Reversing  gears,  439 

Doble,  448 

Hackworth,  446 

Internal-combustion  engine,  453 

Joy,  446 

Marshall,  446 

Stephenson,  440 

Walshaert,  443 

Right-hand  and  left-hand  engine,  14 
Rim  bolts  and  links,  639 
Rope  wheel  grooves,  633 
Run  over  or  under,  15 


S 


Saturated  steam,  66 
Self -oiling  bearing,  112 
Shaft,  568 

application  of  formulas,  605 

at  dead  center,  600 

at  maximum  turning  effort,  602 

center-crank,  587,  598 

collars,  609 

internal-combustion     engine,     596, 
598 

material,  589 

multi-throw,  589 

side-crank,  587,  590 

special  formulas  for,  604 

steam  engine,  590,  598 

steam  turbine,  664 
Shifting  eccentric,  420 
Shock,  468 

Shrouded  blades,  28,  257 
Side-crank  and  center-crank  shafts,  13 
Sight  feed,  111 
Simple  impulse  turbine,  24 
Simple  steam  engine,  18 


INDEX 


703 


Single-acting  and  double-acting  engines, 

13 

Sleeve  motor  gear,  456 
Slider  crank,  5,  283 

efficiency  of,  292 
Speed,  81,  143,  191 

fluctuation  of,  351,  376 
Spring  governors — see  governors. 
Springs,  384,  452 
Standard  engines,  149,  179 
Standards,  689 

Starting  internal-combustion  engines,  41 
Steam,  65 

flow  of,  68 

formation  of,  65 

high  pressure,  91 

jacket,  86 

passages,  266,  503 

superheated,  66,  90,  244 
Steam  consumption,  22,  144,  179,  208 

of  Rankine  cycle,  71,  72 
Steam  cycle,  3 
Steam  engine,  5,  127 

compound,  18,  152 

condensing,  20 

noncondensing,  20 

operation  of,  6 

simple,  18,  127 

uniflow,  21,  135 
Steam  turbine,  23,  206 

bearings,  117 

bleeder,  30 

casing  and  details,  674 

combinations,  29,  260 

compound,  30 

condensing,  31 

conservation  of  residual  velocity  in, 
248 

dimensions,  245 

double-flow,  29 

effect  of  condenser  on,  265 

elements  of,  24 

factors  influencing  operation  of,  264 

impulse-and-reaction,  30 

low-pressure,  30 

mixed-pressure,  30 

noncondensing,  31 

number  of  stages,  246 

pressure-stage  impulse,  26,  234,  268 


Steam    turbine,    pressure-velocity-stage 
impulse,  29,  249,  268 

reaction,  27,  254,  279 

repeated-flow,  29 

shaft,  664 

simple  impulse,  24,  222 

steam  passages  in,  266 

superheated  steam  in,  244 

unequal  wheel  diameters  in,  247 

velocity-stage  impulse,  25,  229 
practical  notes  on,  231 

wheels,  649 

Steam  velocity,  403,  504 
Steel,  casting,  471 

chromium-vanadium,  474 

machinery,  471,  472 

nickel,  472 

nickel-chromium,  472 
Strength  of  materials,  470    • 
Stress,  repeated,  466 

reversed,  466 

static,  466 

suddenly-applied,  468 

working,  464 
Stroke  diagram,  140 
Strut  formula,  485 
Stuffing  box,  506 
Suddenly-applied  load,  468 
Superheated  steam,  66,  90,  244 
Swinging  eccentric,  420 


Temperature  rise  due  to  combustion,  61 

Terminal  drop,  88,  134 

Terminal  pressure,  10 

Three-port  engine,  37 

Throttling  engine,  16,  138 

Thrust  bearing,  119 

Thrust  of  piston,  140,  170,  201 

Thrust,  reversal  of,  304 

Torsion  of  rectangular  shaft,  484 

Total  heat  of  saturated  steam,  67 

Total  ratio  of  expansion,  19 

Trajectory,  250,  278 

Turbine,  bearings,  117 

bearing  temperature,  106 

blade  design,  657 
reaction,  216,  217 


704 


INDEX 


Turbine,  casings,  674 
details,  674 
horsepower,  72,  208 
kilowatts,  209 
shaft,  664 

critical  speed  of,  666 

nature  of  problem,  665 
wheels,  694 

application  of  formulas,  659 

material,  656 

radial  stress,  655 

tangential  stress,  654 
Turning  effort,  289,  310,  314 

diagram,     internal-combustion     en- 
gine, 315,  319-322 

steam  engine,  311,  314 


U 


Uniflow  engine,  21,  135 

U.  S.  Standard  bolts  and  nuts,  695 

Unwin's  flywheel  formulas,  626 


Valve,  mixed-pressure,  398 

double-ported,  419 

operation,  8 

poppet,  405,  408,  459 

slide,  403 

springs,  452 

stems,  Corliss,  436 
slide  valve,  426 
Valve  diagram,  6 

Bilgram,  417 


Valve,  Corliss,  426 

long-range,  432 

rectangular,  422,  457 
Valve  gear,  401 

classification,  408 

cam,  448 

cam-and-eccentric,  438,  452 

Corliss,  426,  436 

details,  457 

high-speed  Corliss,  437 

internal-combustion  engine,  448 

Lentz,  439 

Mclntosh  and  Seymour,  457 

reversing — see  reversing  gears 

single-valve,  410 

sleeve  motor,  456 
Variable  lead,  445 
Velocity  diagrams,   222,   229,   254,   269, 

270,  278 

Velocity  due  to  reheating,  244 
Velocity  of  gas,  95,  407 

of  piston,  287 

of  steam,  403 

relative  and  absolute,  223 
Volumetric  efficiency,  60,  189 

W 

Water  power,  inadequate,  1 

Water  rate — see  steam  consumption 

Wedge  bolts,  555,  617 

Wet  steam,  67 

Wheels,  engine,  350,  623 

turbine,  649 

Work  of  expansion  when  pvn  =  constant, 
64,  128 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  datejsj#mBedJt>elow. 

INSURING 


DEC  13  1948 


LD  21-100m-9,'47(A5702sl6)476 


VC  33320 


414821 


5 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


